diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 19ba83c..fac3ebe 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -48,7 +48,7 @@ jobs:
     strategy:
       fail-fast: false
       matrix:
-        environment: [ci-py310, ci-py313]
+        environment: [ci-py310, ci-py313, ci-backends]
         runs-on: [ubuntu-latest]
 
     steps:
diff --git a/docs/api-reference.md b/docs/api-reference.md
index ffe68f2..b43c960 100644
--- a/docs/api-reference.md
+++ b/docs/api-reference.md
@@ -6,6 +6,7 @@
     :nosignatures:
     :toctree: generated
 
+    at
     atleast_nd
     cov
     create_diagonal
diff --git a/docs/conf.py b/docs/conf.py
index 38f0c8d..79000c9 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -53,6 +53,7 @@
 
 intersphinx_mapping = {
     "python": ("https://docs.python.org/3", None),
+    "jax": ("https://jax.readthedocs.io/en/latest", None),
 }
 
 nitpick_ignore = [
diff --git a/pixi.lock b/pixi.lock
index fe4a855..56a2a9b 100644
--- a/pixi.lock
+++ b/pixi.lock
@@ -1,5 +1,512 @@
 version: 6
 environments:
+  ci-backends:
+    channels:
+    - url: https://prefix.dev/conda-forge/
+    indexes:
+    - https://pypi.org/simple
+    packages:
+      linux-64:
+      - conda: https://prefix.dev/conda-forge/linux-64/_libgcc_mutex-0.1-conda_forge.tar.bz2
+      - conda: https://prefix.dev/conda-forge/linux-64/_openmp_mutex-4.5-2_kmp_llvm.tar.bz2
+      - conda: https://prefix.dev/conda-forge/noarch/array-api-compat-1.10.0-pyhd8ed1ab_0.conda
+      - conda: https://prefix.dev/conda-forge/noarch/array-api-strict-2.2-pyhd8ed1ab_1.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-auth-0.8.0-hb921021_15.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-cal-0.8.1-h1a47875_3.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-common-0.10.6-hb9d3cd8_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-compression-0.3.0-h4e1184b_5.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-event-stream-0.5.0-h7959bf6_11.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-http-0.9.2-hefd7a92_4.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-io-0.15.3-h831e299_5.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-mqtt-0.11.0-h11f4f37_12.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-s3-0.7.7-hf454442_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-c-sdkutils-0.2.1-h4e1184b_4.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-checksums-0.2.2-h4e1184b_4.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-crt-cpp-0.29.7-hd92328a_7.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/aws-sdk-cpp-1.11.458-hc430e4a_4.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/azure-core-cpp-1.14.0-h5cfcd09_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/azure-identity-cpp-1.10.0-h113e628_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/azure-storage-blobs-cpp-12.13.0-h3cf044e_1.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/azure-storage-common-cpp-12.8.0-h736e048_1.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/azure-storage-files-datalake-cpp-12.12.0-ha633028_1.conda
+      - conda: https://prefix.dev/conda-forge/noarch/bokeh-3.6.2-pyhd8ed1ab_1.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/brotli-python-1.1.0-py310hf71b8c6_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/bzip2-1.0.8-h4bc722e_7.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/c-ares-1.34.4-hb9d3cd8_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/ca-certificates-2024.12.14-hbcca054_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/cffi-1.17.1-py310h8deb56e_0.conda
+      - conda: https://prefix.dev/conda-forge/noarch/click-8.1.8-pyh707e725_0.conda
+      - conda: https://prefix.dev/conda-forge/noarch/cloudpickle-3.1.0-pyhd8ed1ab_2.conda
+      - conda: https://prefix.dev/conda-forge/noarch/colorama-0.4.6-pyhd8ed1ab_1.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/contourpy-1.3.1-py310h3788b33_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/coverage-7.6.10-py310h89163eb_0.conda
+      - conda: https://prefix.dev/conda-forge/noarch/cpython-3.10.16-py310hd8ed1ab_1.conda
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+      - conda: https://prefix.dev/conda-forge/linux-64/glog-0.7.1-hbabe93e_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/gmp-6.3.0-hac33072_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/gmpy2-2.1.5-py310he8512ff_3.conda
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+      - conda: https://prefix.dev/conda-forge/linux-64/krb5-1.21.3-h659f571_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/lcms2-2.16-hb7c19ff_0.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/ld_impl_linux-64-2.43-h712a8e2_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/lerc-4.0.0-h27087fc_0.tar.bz2
+      - conda: https://prefix.dev/conda-forge/linux-64/libabseil-20240722.0-cxx17_hbbce691_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libarrow-18.1.0-h44a453e_6_cpu.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libarrow-acero-18.1.0-hcb10f89_6_cpu.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libarrow-dataset-18.1.0-hcb10f89_6_cpu.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libarrow-substrait-18.1.0-h3ee7192_6_cpu.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libblas-3.9.0-26_linux64_openblas.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libbrotlicommon-1.1.0-hb9d3cd8_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libbrotlidec-1.1.0-hb9d3cd8_2.conda
+      - conda: https://prefix.dev/conda-forge/linux-64/libbrotlienc-1.1.0-hb9d3cd8_2.conda
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+- conda: https://prefix.dev/conda-forge/win-64/xorg-libxau-1.0.12-h0e40799_0.conda
+  sha256: 047836241b2712aab1e29474a6f728647bff3ab57de2806b0bb0a6cf9a2d2634
+  md5: 2ffbfae4548098297c033228256eb96e
+  depends:
+  - libgcc >=13
+  - libwinpthread >=12.0.0.r4.gg4f2fc60ca
+  - ucrt >=10.0.20348.0
+  license: MIT
+  license_family: MIT
+  purls: []
+  size: 108013
+  timestamp: 1734229474049
+- conda: https://prefix.dev/conda-forge/linux-64/xorg-libxdmcp-1.1.5-hb9d3cd8_0.conda
+  sha256: 6b250f3e59db07c2514057944a3ea2044d6a8cdde8a47b6497c254520fade1ee
+  md5: 8035c64cb77ed555e3f150b7b3972480
+  depends:
+  - __glibc >=2.17,<3.0.a0
+  - libgcc >=13
+  license: MIT
+  license_family: MIT
+  purls: []
+  size: 19901
+  timestamp: 1727794976192
+- conda: https://prefix.dev/conda-forge/osx-arm64/xorg-libxdmcp-1.1.5-hd74edd7_0.conda
+  sha256: 9939a166d780700d81023546759102b33fdc2c5f11ef09f5f66c77210fd334c8
+  md5: 77c447f48cab5d3a15ac224edb86a968
+  depends:
+  - __osx >=11.0
+  license: MIT
+  license_family: MIT
+  purls: []
+  size: 18487
+  timestamp: 1727795205022
+- conda: https://prefix.dev/conda-forge/win-64/xorg-libxdmcp-1.1.5-h0e40799_0.conda
+  sha256: 9075f98dcaa8e9957e4a3d9d30db05c7578a536950a31c200854c5c34e1edb2c
+  md5: 8393c0f7e7870b4eb45553326f81f0ff
+  depends:
+  - libgcc >=13
+  - libwinpthread >=12.0.0.r4.gg4f2fc60ca
+  - ucrt >=10.0.20348.0
+  license: MIT
+  license_family: MIT
+  purls: []
+  size: 69920
+  timestamp: 1727795651979
+- conda: https://prefix.dev/conda-forge/noarch/xyzservices-2024.9.0-pyhd8ed1ab_1.conda
+  sha256: 5f8757092fc985d7586f2659505ec28757c05fd65d8d6ae549a5cec7e3376977
+  md5: c79cea50b258f652010cb6c8d81591b5
+  depends:
+  - python >=3.8
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/xyzservices?source=hash-mapping
+  size: 46860
+  timestamp: 1733143536777
 - conda: https://prefix.dev/conda-forge/linux-64/yaml-0.2.5-h7f98852_2.tar.bz2
   sha256: a4e34c710eeb26945bdbdaba82d3d74f60a78f54a874ec10d373811a5d217535
   md5: 4cb3ad778ec2d5a7acbdf254eb1c42ae
@@ -4113,6 +9878,28 @@ packages:
   purls: []
   size: 63274
   timestamp: 1641347623319
+- conda: https://prefix.dev/conda-forge/noarch/zict-3.0.0-pyhd8ed1ab_1.conda
+  sha256: 5488542dceeb9f2874e726646548ecc5608060934d6f9ceaa7c6a48c61f9cc8d
+  md5: e52c2ef711ccf31bb7f70ca87d144b9e
+  depends:
+  - python >=3.9
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/zict?source=hash-mapping
+  size: 36341
+  timestamp: 1733261642963
+- conda: https://prefix.dev/conda-forge/noarch/zipp-3.21.0-pyhd8ed1ab_1.conda
+  sha256: 567c04f124525c97a096b65769834b7acb047db24b15a56888a322bf3966c3e1
+  md5: 0c3cc595284c5e8f0f9900a9b228a332
+  depends:
+  - python >=3.9
+  license: MIT
+  license_family: MIT
+  purls:
+  - pkg:pypi/zipp?source=hash-mapping
+  size: 21809
+  timestamp: 1732827613585
 - conda: https://prefix.dev/conda-forge/linux-64/zlib-1.3.1-hb9d3cd8_2.conda
   sha256: 5d7c0e5f0005f74112a34a7425179f4eb6e73c92f5d109e6af4ddeca407c92ab
   md5: c9f075ab2f33b3bbee9e62d4ad0a6cd8
@@ -4136,6 +9923,23 @@ packages:
   purls: []
   size: 77606
   timestamp: 1727963209370
+- conda: https://prefix.dev/conda-forge/linux-64/zstandard-0.23.0-py310ha39cb0e_1.conda
+  sha256: fcd784735205d6c5f19dcb339f92d2eede9bc42a01ec2c384381ee1b6089d4f6
+  md5: f49de34fb99934bf49ab330b5caffd64
+  depends:
+  - __glibc >=2.17,<3.0.a0
+  - cffi >=1.11
+  - libgcc >=13
+  - python >=3.10,<3.11.0a0
+  - python_abi 3.10.* *_cp310
+  - zstd >=1.5.6,<1.5.7.0a0
+  - zstd >=1.5.6,<1.6.0a0
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/zstandard?source=hash-mapping
+  size: 408309
+  timestamp: 1725305719512
 - conda: https://prefix.dev/conda-forge/linux-64/zstandard-0.23.0-py313h80202fe_1.conda
   sha256: ea82f2b8964150a3aa7373b4697e48e64f2200fe68ae554ee85c641c692d1c97
   md5: c178558ff516cd507763ffee230c20b2
@@ -4153,6 +9957,23 @@ packages:
   - pkg:pypi/zstandard?source=hash-mapping
   size: 424424
   timestamp: 1725305749031
+- conda: https://prefix.dev/conda-forge/osx-arm64/zstandard-0.23.0-py310h2665a74_1.conda
+  sha256: a90d06cbfa50fc9b3c37bd092d559452475f22425bacf28f04ecac2e8b1c389c
+  md5: 81b300570a423c9c9521b79f8f2ed1ba
+  depends:
+  - __osx >=11.0
+  - cffi >=1.11
+  - python >=3.10,<3.11.0a0
+  - python >=3.10,<3.11.0a0 *_cpython
+  - python_abi 3.10.* *_cp310
+  - zstd >=1.5.6,<1.5.7.0a0
+  - zstd >=1.5.6,<1.6.0a0
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/zstandard?source=hash-mapping
+  size: 320810
+  timestamp: 1725305704555
 - conda: https://prefix.dev/conda-forge/osx-arm64/zstandard-0.23.0-py313hf2da073_1.conda
   sha256: 12b4e34acff24d291e2626c6610dfd819b8d99a461025ae59affcb6e84bc1d57
   md5: deebca66926691fadaaf16da05ecb5f9
@@ -4170,6 +9991,24 @@ packages:
   - pkg:pypi/zstandard?source=hash-mapping
   size: 336496
   timestamp: 1725305912716
+- conda: https://prefix.dev/conda-forge/win-64/zstandard-0.23.0-py310he5e10e1_1.conda
+  sha256: 4e8aff4d0d42024e9f70783e51666186a681384d59fdd03fafda4b28f1fd540e
+  md5: 2a879227ccc1a10a2caddf12607ffaeb
+  depends:
+  - cffi >=1.11
+  - python >=3.10,<3.11.0a0
+  - python_abi 3.10.* *_cp310
+  - ucrt >=10.0.20348.0
+  - vc >=14.2,<15
+  - vc14_runtime >=14.29.30139
+  - zstd >=1.5.6,<1.5.7.0a0
+  - zstd >=1.5.6,<1.6.0a0
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/zstandard?source=hash-mapping
+  size: 311278
+  timestamp: 1725306039901
 - conda: https://prefix.dev/conda-forge/win-64/zstandard-0.23.0-py313h574b89f_1.conda
   sha256: 1d2744ec0e91da267ce749e19142081472539cb140a7dad0646cd249246691fe
   md5: 8e017aca933f4dd25491151edd3e7820
diff --git a/pyproject.toml b/pyproject.toml
index bac02d2..58c02fa 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -26,7 +26,7 @@ classifiers = [
     "Typing :: Typed",
 ]
 dynamic = ["version"]
-dependencies = ["array-api-compat>=1.1.1"]
+dependencies = ["array-api-compat>=1.10.0,<2"]
 
 [project.optional-dependencies]
 tests = [
@@ -62,8 +62,8 @@ channels = ["https://prefix.dev/conda-forge"]
 platforms = ["linux-64", "osx-arm64", "win-64"]
 
 [tool.pixi.dependencies]
-python = ">=3.10.15,<3.14"
-array-api-compat = ">=1.1.1"
+python = ">=3.10,<3.14"
+array-api-compat = ">=1.10.0,<2"
 
 [tool.pixi.pypi-dependencies]
 array-api-extra = { path = ".", editable = true }
@@ -130,6 +130,35 @@ python = "~=3.10.0"
 [tool.pixi.feature.py313.dependencies]
 python = "~=3.13.0"
 
+# Backends that can run on CPU-only hosts
+[tool.pixi.feature.backends.target.linux-64.dependencies]
+pytorch = "*"
+dask = "*"
+sparse = ">=0.15"
+jax = "*"
+
+[tool.pixi.feature.backends.target.osx-arm64.dependencies]
+pytorch = "*"
+dask = "*"
+sparse = ">=0.15"
+jax = "*"
+
+[tool.pixi.feature.backends.target.win-64.dependencies]
+# pytorch = "*"  # Package unavailable on Windows
+dask = "*"
+sparse = ">=0.15"
+# jax = "*"  # Package unavailable on Windows
+
+# Backends that require a GPU host and a CUDA driver
+[tool.pixi.feature.cuda-backends.target.linux-64.dependencies]
+cupy = "*"
+
+[tool.pixi.feature.cuda-backends.target.osx-arm64.dependencies]
+# cupy = "*"  # Package unavailable on macOSX
+
+[tool.pixi.feature.cuda-backends.target.win-64.dependencies]
+cupy = "*"
+
 [tool.pixi.environments]
 default = { solve-group = "default" }
 lint = { features = ["lint"], solve-group = "default" }
@@ -138,7 +167,9 @@ docs = { features = ["docs"], solve-group = "default" }
 dev = { features = ["lint", "tests", "docs", "dev"], solve-group = "default" }
 ci-py310 = ["py310", "tests"]
 ci-py313 = ["py313", "tests"]
-
+# CUDA not available on free github actions
+ci-backends = ["py310", "tests", "backends"]
+tests-backends = ["py310", "tests", "backends", "cuda-backends"]
 
 # pytest
 
@@ -195,6 +226,8 @@ reportAny = false
 reportExplicitAny = false
 # data-apis/array-api-strict#6
 reportUnknownMemberType = false
+# no array-api-compat type stubs
+reportUnknownVariableType = false
 
 
 # Ruff
@@ -236,6 +269,7 @@ ignore = [
     "PLR09",   # Too many <...>
     "PLR2004", # Magic value used in comparison
     "ISC001",  # Conflicts with formatter
+    "N801",    # Class name should use CapWords convention
     "N802",    # Function name should be lowercase
     "N806",    # Variable in function should be lowercase
 ]
@@ -271,6 +305,7 @@ checks = [
     "ES01",
 ]
 exclude = [ # don't report on objects that match any of these regex
+    '.*test_at.*',
     '.*test_funcs.*',
     '.*test_utils.*',
     '.*test_version.*',
diff --git a/src/array_api_extra/__init__.py b/src/array_api_extra/__init__.py
index 83808e0..09e5ab1 100644
--- a/src/array_api_extra/__init__.py
+++ b/src/array_api_extra/__init__.py
@@ -1,6 +1,7 @@
 """Extra array functions built on top of the array API standard."""
 
 from ._funcs import (
+    at,
     atleast_nd,
     cov,
     create_diagonal,
@@ -16,6 +17,7 @@
 # pylint: disable=duplicate-code
 __all__ = [
     "__version__",
+    "at",
     "atleast_nd",
     "cov",
     "create_diagonal",
diff --git a/src/array_api_extra/_funcs.py b/src/array_api_extra/_funcs.py
index 369319e..03b4829 100644
--- a/src/array_api_extra/_funcs.py
+++ b/src/array_api_extra/_funcs.py
@@ -1,12 +1,22 @@
 """Public API Functions."""
 
+import operator
 import warnings
 
+# https://github.com/pylint-dev/pylint/issues/10112
+from collections.abc import Callable  # pylint: disable=import-error
+from typing import ClassVar, Literal
+
 from ._lib import _compat, _utils
-from ._lib._compat import array_namespace
-from ._lib._typing import Array, ModuleType
+from ._lib._compat import (
+    array_namespace,
+    is_jax_array,
+    is_writeable_array,
+)
+from ._lib._typing import Array, Index, ModuleType
 
 __all__ = [
+    "at",
     "atleast_nd",
     "cov",
     "create_diagonal",
@@ -589,3 +599,323 @@ def pad(
     )
     padded[(slice(pad_width, -pad_width, None),) * x.ndim] = x
     return padded
+
+
+_undef = object()
+
+
+class at:  # pylint: disable=invalid-name  # numpydoc ignore=PR02
+    """
+    Update operations for read-only arrays.
+
+    This implements ``jax.numpy.ndarray.at`` for all writeable
+    backends (those that support ``__setitem__``) and routes
+    to the ``.at[]`` method for JAX arrays.
+
+    Parameters
+    ----------
+    x : array
+        Input array.
+    idx : index, optional
+        Only `array API standard compliant indices
+        <https://data-apis.org/array-api/latest/API_specification/indexing.html>`_
+        are supported.
+
+        You may use two alternate syntaxes::
+
+          >>> import array_api_extra as xpx
+          >>> xpx.at(x, idx).set(value)  # or add(value), etc.
+          >>> xpx.at(x)[idx].set(value)
+
+    copy : bool, optional
+        None (default)
+            The array parameter *may* be modified in place if it is
+            possible and beneficial for performance.
+            You should not reuse it after calling this function.
+        True
+            Ensure that the inputs are not modified.
+        False
+            Ensure that the update operation writes back to the input.
+            Raise ``ValueError`` if a copy cannot be avoided.
+
+    xp : array_namespace, optional
+        The standard-compatible namespace for `x`. Default: infer.
+
+    Returns
+    -------
+    Updated input array.
+
+    Warnings
+    --------
+    (a) When you omit the ``copy`` parameter, you should always immediately overwrite
+    the parameter array::
+
+        >>> import array_api_extra as xpx
+        >>> x = xpx.at(x, 0).set(2)
+
+    The anti-pattern below must be avoided, as it will result in different
+    behaviour on read-only versus writeable arrays::
+
+        >>> x = xp.asarray([0, 0, 0])
+        >>> y = xpx.at(x, 0).set(2)
+        >>> z = xpx.at(x, 1).set(3)
+
+    In the above example, ``x == [0, 0, 0]``, ``y == [2, 0, 0]`` and z == ``[0, 3, 0]``
+    when ``x`` is read-only, whereas ``x == y == z == [2, 3, 0]`` when ``x`` is
+    writeable!
+
+    (b) The array API standard does not support integer array indices.
+    The behaviour of update methods when the index is an array of integers is
+    undefined and will vary between backends; this is particularly true when the
+    index contains multiple occurrences of the same index, e.g.::
+
+        >>> import numpy as np
+        >>> import jax.numpy as jnp
+        >>> import array_api_extra as xpx
+        >>> xpx.at(np.asarray([123]), np.asarray([0, 0])).add(1)
+        array([124])
+        >>> xpx.at(jnp.asarray([123]), jnp.asarray([0, 0])).add(1)
+        Array([125], dtype=int32)
+
+    See Also
+    --------
+    jax.numpy.ndarray.at : Equivalent array method in JAX.
+
+    Notes
+    -----
+    `sparse <https://sparse.pydata.org/>`_, as well as read-only arrays from libraries
+    not explicitly covered by ``array-api-compat``, are not supported by update
+    methods.
+
+    Examples
+    --------
+    Given either of these equivalent expressions::
+
+      >>> import array_api_extra as xpx
+      >>> x = xpx.at(x)[1].add(2)
+      >>> x = xpx.at(x, 1).add(2)
+
+    If x is a JAX array, they are the same as::
+
+      >>> x = x.at[1].add(2)
+
+    If x is a read-only numpy array, they are the same as::
+
+      >>> x = x.copy()
+      >>> x[1] += 2
+
+    For other known backends, they are the same as::
+
+      >>> x[1] += 2
+    """
+
+    _x: Array
+    _idx: Index
+    __slots__: ClassVar[tuple[str, ...]] = ("_idx", "_x")
+
+    def __init__(
+        self, x: Array, idx: Index = _undef, /
+    ) -> None:  # numpydoc ignore=GL08
+        self._x = x
+        self._idx = idx
+
+    def __getitem__(self, idx: Index, /) -> "at":  # numpydoc ignore=PR01,RT01
+        """
+        Allow for the alternate syntax ``at(x)[start:stop:step]``.
+
+        It looks prettier than ``at(x, slice(start, stop, step))``
+        and feels more intuitive coming from the JAX documentation.
+        """
+        if self._idx is not _undef:
+            msg = "Index has already been set"
+            raise ValueError(msg)
+        return at(self._x, idx)
+
+    def _update_common(
+        self,
+        at_op: str,
+        y: Array,
+        /,
+        copy: bool | None,
+        xp: ModuleType | None,
+    ) -> tuple[Array, None] | tuple[None, Array]:  # numpydoc ignore=PR01
+        """
+        Perform common prepocessing to all update operations.
+
+        Returns
+        -------
+        tuple
+            If the operation can be resolved by ``at[]``, ``(return value, None)``
+            Otherwise, ``(None, preprocessed x)``.
+        """
+        x, idx = self._x, self._idx
+
+        if idx is _undef:
+            msg = (
+                "Index has not been set.\n"
+                "Usage: either\n"
+                "    at(x, idx).set(value)\n"
+                "or\n"
+                "    at(x)[idx].set(value)\n"
+                "(same for all other methods)."
+            )
+            raise ValueError(msg)
+
+        if copy not in (True, False, None):
+            msg = f"copy must be True, False, or None; got {copy!r}"  # pyright: ignore[reportUnreachable]
+            raise ValueError(msg)
+
+        if copy is None:
+            writeable = is_writeable_array(x)
+            copy = not writeable
+        elif copy:
+            writeable = None
+        else:
+            writeable = is_writeable_array(x)
+
+        if copy:
+            if is_jax_array(x):
+                # Use JAX's at[]
+                func = getattr(x.at[idx], at_op)
+                return func(y), None
+            # Emulate at[] behaviour for non-JAX arrays
+            # with a copy followed by an update
+            if xp is None:
+                xp = array_namespace(x)
+            x = xp.asarray(x, copy=True)
+            if writeable is False:
+                # A copy of a read-only numpy array is writeable
+                # Note: this assumes that a copy of a writeable array is writeable
+                writeable = None
+
+        if writeable is None:
+            writeable = is_writeable_array(x)
+        if not writeable:
+            # sparse crashes here
+            msg = f"Can't update read-only array {x}"
+            raise ValueError(msg)
+
+        return None, x
+
+    def set(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] = y`` and return the update array."""
+        res, x = self._update_common("set", y, copy=copy, xp=xp)
+        if res is not None:
+            return res
+        assert x is not None
+        x[self._idx] = y
+        return x
+
+    def _iop(
+        self,
+        at_op: Literal[
+            "set", "add", "subtract", "multiply", "divide", "power", "min", "max"
+        ],
+        elwise_op: Callable[[Array, Array], Array],
+        y: Array,
+        /,
+        copy: bool | None,
+        xp: ModuleType | None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """
+        ``x[idx] += y`` or equivalent in-place operation on a subset of x.
+
+        which is the same as saying
+            x[idx] = x[idx] + y
+        Note that this is not the same as
+            operator.iadd(x[idx], y)
+        Consider for example when x is a numpy array and idx is a fancy index, which
+        triggers a deep copy on __getitem__.
+        """
+        res, x = self._update_common(at_op, y, copy=copy, xp=xp)
+        if res is not None:
+            return res
+        assert x is not None
+        x[self._idx] = elwise_op(x[self._idx], y)
+        return x
+
+    def add(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] += y`` and return the updated array."""
+
+        # Note for this and all other methods based on _iop:
+        # operator.iadd and operator.add subtly differ in behaviour, as
+        # only iadd will trigger exceptions when y has an incompatible dtype.
+        return self._iop("add", operator.iadd, y, copy=copy, xp=xp)
+
+    def subtract(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] -= y`` and return the updated array."""
+        return self._iop("subtract", operator.isub, y, copy=copy, xp=xp)
+
+    def multiply(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] *= y`` and return the updated array."""
+        return self._iop("multiply", operator.imul, y, copy=copy, xp=xp)
+
+    def divide(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] /= y`` and return the updated array."""
+        return self._iop("divide", operator.itruediv, y, copy=copy, xp=xp)
+
+    def power(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] **= y`` and return the updated array."""
+        return self._iop("power", operator.ipow, y, copy=copy, xp=xp)
+
+    def min(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] = minimum(x[idx], y)`` and return the updated array."""
+        if xp is None:
+            xp = array_namespace(self._x)
+        y = xp.asarray(y)
+        return self._iop("min", xp.minimum, y, copy=copy, xp=xp)
+
+    def max(
+        self,
+        y: Array,
+        /,
+        copy: bool | None = None,
+        xp: ModuleType | None = None,
+    ) -> Array:  # numpydoc ignore=PR01,RT01
+        """Apply ``x[idx] = maximum(x[idx], y)`` and return the updated array."""
+        if xp is None:
+            xp = array_namespace(self._x)
+        y = xp.asarray(y)
+        return self._iop("max", xp.maximum, y, copy=copy, xp=xp)
diff --git a/src/array_api_extra/_lib/_compat.py b/src/array_api_extra/_lib/_compat.py
index de7a220..a24175d 100644
--- a/src/array_api_extra/_lib/_compat.py
+++ b/src/array_api_extra/_lib/_compat.py
@@ -4,16 +4,22 @@
 
 try:
     from ..._array_api_compat_vendor import (  # pyright: ignore[reportMissingImports]
-        array_namespace,  # pyright: ignore[reportUnknownVariableType]
-        device,  # pyright: ignore[reportUnknownVariableType]
+        array_namespace,
+        device,
+        is_jax_array,
+        is_writeable_array,
     )
 except ImportError:
     from array_api_compat import (  # pyright: ignore[reportMissingTypeStubs]
-        array_namespace,  # pyright: ignore[reportUnknownVariableType]
+        array_namespace,
         device,
+        is_jax_array,
+        is_writeable_array,
     )
 
 __all__ = [
     "array_namespace",
     "device",
+    "is_jax_array",
+    "is_writeable_array",
 ]
diff --git a/src/array_api_extra/_lib/_compat.pyi b/src/array_api_extra/_lib/_compat.pyi
index 2105708..f65a28f 100644
--- a/src/array_api_extra/_lib/_compat.pyi
+++ b/src/array_api_extra/_lib/_compat.pyi
@@ -6,12 +6,14 @@ from ._typing import Array, Device
 
 # pylint: disable=missing-class-docstring,unused-argument
 
-class ArrayModule(ModuleType):  # numpydoc ignore=GL08
-    def device(self, x: Array, /) -> Device: ...  # numpydoc ignore=GL08
+class ArrayModule(ModuleType):
+    def device(self, x: Array, /) -> Device: ...
 
 def array_namespace(
     *xs: Array,
     api_version: str | None = None,
     use_compat: bool | None = None,
-) -> ArrayModule: ...  # numpydoc ignore=GL08
-def device(x: Array, /) -> Device: ...  # numpydoc ignore=GL08
+) -> ArrayModule: ...
+def device(x: Array, /) -> Device: ...
+def is_jax_array(x: object, /) -> bool: ...
+def is_writeable_array(x: object, /) -> bool: ...
diff --git a/src/array_api_extra/_lib/_typing.py b/src/array_api_extra/_lib/_typing.py
index 8aedc44..c612020 100644
--- a/src/array_api_extra/_lib/_typing.py
+++ b/src/array_api_extra/_lib/_typing.py
@@ -6,5 +6,6 @@
 # To be changed to a Protocol later (see data-apis/array-api#589)
 Array = Any  # type: ignore[no-any-explicit]
 Device = Any  # type: ignore[no-any-explicit]
+Index = Any  # type: ignore[no-any-explicit]
 
-__all__ = ["Array", "Device", "ModuleType"]
+__all__ = ["Array", "Device", "Index", "ModuleType"]
diff --git a/tests/test_at.py b/tests/test_at.py
new file mode 100644
index 0000000..1f66067
--- /dev/null
+++ b/tests/test_at.py
@@ -0,0 +1,158 @@
+from collections.abc import Generator
+from contextlib import contextmanager
+from importlib import import_module
+
+import numpy as np
+import pytest
+from array_api_compat import (  # type: ignore[import-untyped]  # pyright: ignore[reportMissingTypeStubs]
+    array_namespace,
+    is_dask_array,
+    is_pydata_sparse_array,
+    is_writeable_array,
+)
+
+from array_api_extra import at
+from array_api_extra._lib._typing import Array
+
+all_libraries = (
+    "array_api_strict",
+    "numpy",
+    "numpy_readonly",
+    "cupy",
+    "torch",
+    "dask.array",
+    "sparse",
+    "jax.numpy",
+)
+
+
+@pytest.fixture(params=all_libraries)
+def array(request: pytest.FixtureRequest) -> Array:
+    library = request.param
+    if library == "numpy_readonly":
+        x = np.asarray([10.0, 20.0, 30.0])
+        x.flags.writeable = False
+    else:
+        try:
+            lib = import_module(library)
+        except ImportError:
+            pytest.skip(f"{library} is not installed")
+        x = lib.asarray([10.0, 20.0, 30.0])
+    return x
+
+
+def assert_array_equal(a: Array, b: Array) -> None:
+    xp = array_namespace(a)
+    b = xp.asarray(b)
+    eq = xp.all(a == b)
+    if is_dask_array(a):
+        eq = eq.compute()
+    assert eq
+
+
+@contextmanager
+def assert_copy(array: Array, copy: bool | None) -> Generator[None, None, None]:
+    if copy is False and not is_writeable_array(array):
+        with pytest.raises((TypeError, ValueError)):
+            yield
+        return
+
+    xp = array_namespace(array)
+    array_orig = xp.asarray(array, copy=True)
+    yield
+
+    if copy is None:
+        copy = not is_writeable_array(array)
+    assert_array_equal(xp.all(array == array_orig), copy)
+
+
+@pytest.mark.parametrize(
+    ("kwargs", "expect_copy"),
+    [
+        ({"copy": True}, True),
+        ({"copy": False}, False),
+        ({"copy": None}, None),  # Behavior is backend-specific
+        ({}, None),  # Test that the copy parameter defaults to None
+    ],
+)
+@pytest.mark.parametrize(
+    ("op", "arg", "expect"),
+    [
+        ("set", 40.0, [10.0, 40.0, 40.0]),
+        ("add", 40.0, [10.0, 60.0, 70.0]),
+        ("subtract", 100.0, [10.0, -80.0, -70.0]),
+        ("multiply", 2.0, [10.0, 40.0, 60.0]),
+        ("divide", 2.0, [10.0, 10.0, 15.0]),
+        ("power", 2.0, [10.0, 400.0, 900.0]),
+        ("min", 25.0, [10.0, 20.0, 25.0]),
+        ("max", 25.0, [10.0, 25.0, 30.0]),
+    ],
+)
+def test_update_ops(
+    array: Array,
+    kwargs: dict[str, bool | None],
+    expect_copy: bool | None,
+    op: str,
+    arg: float,
+    expect: list[float],
+):
+    if is_pydata_sparse_array(array):
+        pytest.skip("at() does not support updates on sparse arrays")
+
+    with assert_copy(array, expect_copy):
+        y = getattr(at(array)[1:], op)(arg, **kwargs)
+        assert isinstance(y, type(array))
+        assert_array_equal(y, expect)
+
+
+def test_copy_invalid():
+    a = np.asarray([1, 2, 3])
+    with pytest.raises(ValueError, match="copy"):
+        at(a, 0).set(4, copy="invalid")  # type: ignore[arg-type]  # pyright: ignore[reportArgumentType]
+
+
+def test_xp():
+    a = np.asarray([1, 2, 3])
+    at(a, 0).set(4, xp=np)
+    at(a, 0).add(4, xp=np)
+    at(a, 0).subtract(4, xp=np)
+    at(a, 0).multiply(4, xp=np)
+    at(a, 0).divide(4, xp=np)
+    at(a, 0).power(4, xp=np)
+    at(a, 0).min(4, xp=np)
+    at(a, 0).max(4, xp=np)
+
+
+def test_alternate_index_syntax():
+    a = np.asarray([1, 2, 3])
+    assert_array_equal(at(a, 0).set(4, copy=True), [4, 2, 3])
+    assert_array_equal(at(a)[0].set(4, copy=True), [4, 2, 3])
+
+    a_at = at(a)
+    assert_array_equal(a_at[0].add(1, copy=True), [2, 2, 3])
+    assert_array_equal(a_at[1].add(2, copy=True), [1, 4, 3])
+
+    with pytest.raises(ValueError, match="Index"):
+        at(a).set(4)
+    with pytest.raises(ValueError, match="Index"):
+        at(a, 0)[0].set(4)
+
+
+@pytest.mark.parametrize("copy", [True, False])
+@pytest.mark.parametrize("op", ["add", "subtract", "multiply", "divide", "power"])
+def test_iops_incompatible_dtype(op: str, copy: bool):
+    """Test that at() replicates the backend's behaviour for
+    in-place operations with incompatible dtypes.
+
+    Note:
+    >>> a = np.asarray([1, 2, 3])
+    >>> a / 1.5
+    array([0.        , 0.66666667, 1.33333333])
+    >>> a /= 1.5
+    UFuncTypeError: Cannot cast ufunc 'divide' output from dtype('float64')
+    to dtype('int64') with casting rule 'same_kind'
+    """
+    a = np.asarray([2, 4])
+    func = getattr(at(a)[:], op)
+    with pytest.raises(TypeError, match="Cannot cast ufunc"):
+        func(1.1, copy=copy)
diff --git a/vendor_tests/test_vendor.py b/vendor_tests/test_vendor.py
index a248abc..c2e6570 100644
--- a/vendor_tests/test_vendor.py
+++ b/vendor_tests/test_vendor.py
@@ -3,10 +3,18 @@
 
 
 def test_vendor_compat():
-    from ._array_api_compat_vendor import array_namespace
+    from ._array_api_compat_vendor import (  # type: ignore[attr-defined]
+        array_namespace,
+        device,
+        is_jax_array,
+        is_writeable_array,
+    )
 
     x = xp.asarray([1, 2, 3])
     assert array_namespace(x) is xp
+    device(x)
+    assert not is_jax_array(x)
+    assert is_writeable_array(x)
 
 
 def test_vendor_extra():