diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index 39c61a5b..3b58a86d 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -12,8 +12,9 @@ jobs:
     strategy:
       matrix:
         os: [Ubuntu]
-        python-version: ["3.7", "3.8", "3.9", "3.10"]
-        sphinx-version: ["sphinx==3.0", "sphinx==4.0", "sphinx>4.4"]
+        python-version: ["3.7", "3.8", "3.9"]
+        sphinx-version:
+          ["sphinx==4.0.2", "sphinx==4.5", "sphinx==5.0", "sphinx>=5.0"]
     steps:
       - uses: actions/checkout@v3
 
@@ -30,10 +31,6 @@ jobs:
           python -m pip install ${{ matrix.sphinx-version }}
           python -m pip list
 
-      - name: Downgrade Jinja2 for sphinx<4
-        if: ${{ matrix.sphinx-version }} == 'sphinx<4.0.2'
-        run: python -m pip install jinja2==3.0.3 markupsafe==2.0.1 pydata-sphinx-theme==0.8.0
-
       - name: Install
         run: |
           python -m pip install .
@@ -64,13 +61,67 @@ jobs:
           make -C doc html SPHINXOPTS="-nT"
           make -C doc latexpdf SPHINXOPTS="-nT"
 
+  test-new:
+    runs-on: ${{ matrix.os }}-latest
+    strategy:
+      matrix:
+        os: [Ubuntu]
+        python-version: ["3.10"]
+        sphinx-version:
+          ["sphinx==4.2", "sphinx==4.5", "sphinx==5.0", "sphinx>=5.0"]
+    steps:
+      - uses: actions/checkout@v3
+
+      - name: Python setup
+        uses: actions/setup-python@v3
+        with:
+          python-version: ${{ matrix.python-version }}
+
+      - name: Setup environment
+        run: |
+          python -m pip install --upgrade pip wheel setuptools
+          python -m pip install -r requirements/test.txt -r doc/requirements.txt
+          python -m pip install codecov
+          python -m pip install ${{ matrix.sphinx-version }}
+          python -m pip list
+
+      - name: Install
+        run: |
+          python -m pip install .
+          pip list
+
+      - name: Run test suite
+        run: |
+          pytest -v --pyargs .
+
+      - name: Test coverage
+        run: |
+          codecov
+
+      - name: Make sure CLI works
+        run: |
+          python -m numpydoc numpydoc.tests.test_main._capture_stdout
+          echo '! python -m numpydoc numpydoc.tests.test_main._invalid_docstring' | bash
+          python -m numpydoc --validate numpydoc.tests.test_main._capture_stdout
+          echo '! python -m numpydoc --validate numpydoc.tests.test_main._docstring_with_errors' | bash
+
+      - name: Setup for doc build
+        run: |
+          sudo apt-get update
+          sudo apt install texlive texlive-latex-extra latexmk dvipng
+
+      - name: Build documentation
+        run: |
+          make -C doc html SPHINXOPTS="-nT"
+          make -C doc latexpdf SPHINXOPTS="-nT"
   base:
     runs-on: ${{ matrix.os }}-latest
     strategy:
       matrix:
         os: [ubuntu, macos, windows]
         python-version: ["3.11-dev"]
-        sphinx-version: ["sphinx==4.0", "sphinx==4.5"]
+        sphinx-version:
+          ["sphinx==4.2", "sphinx==4.5", "sphinx==5.0", "sphinx>=5.0"]
     steps:
       - uses: actions/checkout@v3
 
@@ -86,10 +137,6 @@ jobs:
           python -m pip install ${{ matrix.sphinx-version }}
           python -m pip list
 
-      - name: Downgrade Jinja2 for sphinx<4
-        if: ${{ matrix.sphinx-version }} == 'sphinx<4.0.2'
-        run: python -m pip install jinja2==3.0.3 markupsafe==2.0.1 pydata-sphinx-theme==0.8.0
-
       - name: Install
         run: |
           python -m pip install .
diff --git a/README.rst b/README.rst
index 7d6c34e5..631827a2 100644
--- a/README.rst
+++ b/README.rst
@@ -18,7 +18,7 @@ docstrings formatted according to the NumPy documentation format.
 The extension also adds the code description directives
 ``np:function``, ``np-c:function``, etc.
 
-numpydoc requires Python 3.7+ and sphinx 3.0+.
+numpydoc requires Python 3.7+ and sphinx 4.0.2+.
 
 For usage information, please refer to the `documentation
 <https://numpydoc.readthedocs.io/>`_.
diff --git a/doc/install.rst b/doc/install.rst
index fb3c26bf..939decd5 100644
--- a/doc/install.rst
+++ b/doc/install.rst
@@ -5,7 +5,7 @@ Getting started
 Installation
 ============
 
-This extension requires Python 3.7+, sphinx 3.0+ and is available from:
+This extension requires Python 3.7+, sphinx 4.0.2+ and is available from:
 
 * `numpydoc on PyPI <http://pypi.python.org/pypi/numpydoc>`_
 * `numpydoc on GitHub <https://github.com/numpy/numpydoc/>`_
diff --git a/numpydoc/__init__.py b/numpydoc/__init__.py
index 31cc4249..97f53053 100644
--- a/numpydoc/__init__.py
+++ b/numpydoc/__init__.py
@@ -5,32 +5,6 @@
 from ._version import __version__
 
 
-def _verify_sphinx_jinja():
-    """Ensure sphinx and jinja versions are compatible.
-
-    Jinja2>=3.1 requires Sphinx>=4.0.2. Raises exception if this condition is
-    not met.
-
-    TODO: This check can be removed when the minimum supported sphinx version
-    for numpydoc sphinx>=4.0.2
-    """
-    import sphinx, jinja2
-    from packaging import version
-
-    if version.parse(sphinx.__version__) <= version.parse("4.0.2"):
-        if version.parse(jinja2.__version__) >= version.parse("3.1"):
-            from sphinx.errors import VersionRequirementError
-
-            raise VersionRequirementError(
-                "\n\nSphinx<4.0.2 is incompatible with Jinja2>=3.1.\n"
-                "If you wish to continue using sphinx<4.0.2 you need to pin "
-                "Jinja2<3.1."
-            )
-
-
-_verify_sphinx_jinja()
-
-
 def setup(app, *args, **kwargs):
     from .numpydoc import setup
 
diff --git a/setup.py b/setup.py
index 1b292c38..ac9a9a30 100644
--- a/setup.py
+++ b/setup.py
@@ -52,7 +52,7 @@ def read(fname):
     author_email="pav@iki.fi",
     url="https://numpydoc.readthedocs.io",
     license="BSD",
-    install_requires=["sphinx>=3.0", "Jinja2>=2.10"],
+    install_requires=["sphinx>=4.0.2", "Jinja2>=2.10"],
     python_requires=">=3.7",
     extras_require={
         "testing": [