diff --git a/.github/workflows/cibuildwheel.yml b/.github/workflows/cibuildwheel.yml
index 1c4076d6bb3a..84951cac4784 100644
--- a/.github/workflows/cibuildwheel.yml
+++ b/.github/workflows/cibuildwheel.yml
@@ -47,7 +47,7 @@ jobs:
 
       - name: Install cibuildwheel
         run: |
-          python -m pip install cibuildwheel==1.9.0
+          python -m pip install cibuildwheel==2.1.1
 
       - name: Build minimum NumPy for aarch64
         if: matrix.cibw_archs == 'aarch64' && steps.numpy-cache.outputs.cache-hit != 'true'
@@ -93,7 +93,7 @@ jobs:
           CIBW_BUILD: "pp37-*"
           CIBW_BEFORE_BUILD: pip install certifi numpy==${{ env.min-numpy-version }}
           CIBW_ARCHS: ${{ matrix.cibw_archs }}
-        if: runner.os != 'Windows' && matrix.cibw_archs != 'aarch64'
+        if: matrix.cibw_archs != 'aarch64'
 
       - name: Validate that LICENSE files are included in wheels
         run: |
diff --git a/lib/matplotlib/backend_bases.py b/lib/matplotlib/backend_bases.py
index d7be356d66d2..00e99c9f073e 100644
--- a/lib/matplotlib/backend_bases.py
+++ b/lib/matplotlib/backend_bases.py
@@ -1618,14 +1618,20 @@ def wrapper(*args, **kwargs):
             if frame is None:
                 # when called in embedded context may hit frame is None.
                 break
+            # Work around sphinx-gallery not setting __name__.
+            frame_name = frame.f_globals.get('__name__', '')
             if re.match(r'\A(matplotlib|mpl_toolkits)(\Z|\.(?!tests\.))',
-                        # Work around sphinx-gallery not setting __name__.
-                        frame.f_globals.get('__name__', '')):
-                if public_api.match(frame.f_code.co_name):
-                    name = frame.f_code.co_name
+                        frame_name):
+                name = frame.f_code.co_name
+                if public_api.match(name):
                     if name in ('print_figure', '_no_output_draw'):
                         seen_print_figure = True
 
+            elif frame_name == '_functools':
+                # PyPy adds an extra frame without module prefix for this
+                # functools wrapper, which we ignore to assume we're still in
+                # Matplotlib code.
+                continue
             else:
                 break
 
diff --git a/lib/matplotlib/patches.py b/lib/matplotlib/patches.py
index 3fbf4ee245f7..5e053cb364a3 100644
--- a/lib/matplotlib/patches.py
+++ b/lib/matplotlib/patches.py
@@ -1451,7 +1451,8 @@ def _make_verts(self):
 
 
 docstring.interpd.update(
-    FancyArrow="\n".join(inspect.getdoc(FancyArrow.__init__).splitlines()[2:]))
+    FancyArrow="\n".join(
+        (inspect.getdoc(FancyArrow.__init__) or "").splitlines()[2:]))
 
 
 class CirclePolygon(RegularPolygon):
diff --git a/lib/matplotlib/scale.py b/lib/matplotlib/scale.py
index 180906106747..306a60aa4e66 100644
--- a/lib/matplotlib/scale.py
+++ b/lib/matplotlib/scale.py
@@ -620,10 +620,11 @@ def _get_scale_docs():
     """
     docs = []
     for name, scale_class in _scale_mapping.items():
+        docstring = inspect.getdoc(scale_class.__init__) or ""
         docs.extend([
             f"    {name!r}",
             "",
-            textwrap.indent(inspect.getdoc(scale_class.__init__), " " * 8),
+            textwrap.indent(docstring, " " * 8),
             ""
         ])
     return "\n".join(docs)
diff --git a/lib/matplotlib/tests/test_animation.py b/lib/matplotlib/tests/test_animation.py
index fbd984f70543..a7ba927494b1 100644
--- a/lib/matplotlib/tests/test_animation.py
+++ b/lib/matplotlib/tests/test_animation.py
@@ -1,6 +1,6 @@
-import gc
 import os
 from pathlib import Path
+import platform
 import subprocess
 import sys
 import weakref
@@ -87,10 +87,15 @@ def test_null_movie_writer(anim):
 
 @pytest.mark.parametrize('anim', [dict(klass=dict)], indirect=['anim'])
 def test_animation_delete(anim):
+    if platform.python_implementation() == 'PyPy':
+        # Something in the test setup fixture lingers around into the test and
+        # breaks pytest.warns on PyPy. This garbage collection fixes it.
+        # https://foss.heptapod.net/pypy/pypy/-/issues/3536
+        np.testing.break_cycles()
     anim = animation.FuncAnimation(**anim)
     with pytest.warns(Warning, match='Animation was deleted'):
         del anim
-        gc.collect()
+        np.testing.break_cycles()
 
 
 def test_movie_writer_dpi_default():
@@ -201,6 +206,11 @@ def test_save_animation_smoketest(tmpdir, writer, frame_format, output, anim):
 ])
 @pytest.mark.parametrize('anim', [dict(klass=dict)], indirect=['anim'])
 def test_animation_repr_html(writer, html, want, anim):
+    if platform.python_implementation() == 'PyPy':
+        # Something in the test setup fixture lingers around into the test and
+        # breaks pytest.warns on PyPy. This garbage collection fixes it.
+        # https://foss.heptapod.net/pypy/pypy/-/issues/3536
+        np.testing.break_cycles()
     if (writer == 'imagemagick' and html == 'html5'
             # ImageMagick delegates to ffmpeg for this format.
             and not animation.FFMpegWriter.isAvailable()):
@@ -214,7 +224,8 @@ def test_animation_repr_html(writer, html, want, anim):
     if want is None:
         assert html is None
         with pytest.warns(UserWarning):
-            del anim  # Animtion was never run, so will warn on cleanup.
+            del anim  # Animation was never run, so will warn on cleanup.
+            np.testing.break_cycles()
     else:
         assert want in html
 
@@ -324,6 +335,7 @@ def frames_generator():
     writer = NullMovieWriter()
     anim.save('unused.null', writer=writer)
     assert len(frames_generated) == 5
+    np.testing.break_cycles()
     for f in frames_generated:
         # If cache_frame_data is True, then the weakref should be alive;
         # if cache_frame_data is False, then the weakref should be dead (None).
diff --git a/lib/matplotlib/tests/test_backend_tk.py b/lib/matplotlib/tests/test_backend_tk.py
index 771d5193e397..5218d2c776a3 100644
--- a/lib/matplotlib/tests/test_backend_tk.py
+++ b/lib/matplotlib/tests/test_backend_tk.py
@@ -1,6 +1,7 @@
 import functools
 import inspect
 import os
+import platform
 import re
 import subprocess
 import sys
@@ -111,6 +112,9 @@ def legitimate_quit():
         print("success")
 
 
+@pytest.mark.skipif(platform.python_implementation() != 'CPython',
+                    reason='PyPy does not support Tkinter threading: '
+                           'https://foss.heptapod.net/pypy/pypy/-/issues/1929')
 @pytest.mark.backend('TkAgg', skip_on_importerror=True)
 @pytest.mark.flaky(reruns=3)
 @_isolated_tk_test(success_count=1)
diff --git a/lib/matplotlib/tests/test_backends_interactive.py b/lib/matplotlib/tests/test_backends_interactive.py
index d3e04e986b0c..36ec3bfc83c3 100644
--- a/lib/matplotlib/tests/test_backends_interactive.py
+++ b/lib/matplotlib/tests/test_backends_interactive.py
@@ -3,6 +3,7 @@
 import inspect
 import json
 import os
+import platform
 import signal
 import subprocess
 import sys
@@ -220,6 +221,12 @@ def _test_thread_impl():
     elif param.values[0].get("QT_API") == "PySide2":
         param.marks.append(
             pytest.mark.xfail(raises=subprocess.CalledProcessError))
+    elif backend == "tkagg" and platform.python_implementation() != 'CPython':
+        param.marks.append(
+            pytest.mark.xfail(
+                reason='PyPy does not support Tkinter threading: '
+                       'https://foss.heptapod.net/pypy/pypy/-/issues/1929',
+                strict=True))
 
 
 @pytest.mark.parametrize("env", _thread_safe_backends)
diff --git a/lib/matplotlib/tests/test_cbook.py b/lib/matplotlib/tests/test_cbook.py
index 474ea1a41d93..88fac2dee435 100644
--- a/lib/matplotlib/tests/test_cbook.py
+++ b/lib/matplotlib/tests/test_cbook.py
@@ -198,11 +198,13 @@ def count(self):
         return count1
 
     def is_empty(self):
+        np.testing.break_cycles()
         assert self.callbacks._func_cid_map == {}
         assert self.callbacks.callbacks == {}
         assert self.callbacks._pickled_cids == set()
 
     def is_not_empty(self):
+        np.testing.break_cycles()
         assert self.callbacks._func_cid_map != {}
         assert self.callbacks.callbacks != {}
 
diff --git a/lib/matplotlib/tests/test_quiver.py b/lib/matplotlib/tests/test_quiver.py
index d7a848f61bcc..ca0bf19de5e7 100644
--- a/lib/matplotlib/tests/test_quiver.py
+++ b/lib/matplotlib/tests/test_quiver.py
@@ -1,6 +1,9 @@
+import platform
+import sys
+
 import numpy as np
 import pytest
-import sys
+
 from matplotlib import pyplot as plt
 from matplotlib.testing.decorators import image_comparison
 
@@ -15,6 +18,8 @@ def draw_quiver(ax, **kw):
     return Q
 
 
+@pytest.mark.skipif(platform.python_implementation() != 'CPython',
+                    reason='Requires CPython')
 def test_quiver_memory_leak():
     fig, ax = plt.subplots()
 
@@ -27,6 +32,8 @@ def test_quiver_memory_leak():
     assert sys.getrefcount(ttX) == 2
 
 
+@pytest.mark.skipif(platform.python_implementation() != 'CPython',
+                    reason='Requires CPython')
 def test_quiver_key_memory_leak():
     fig, ax = plt.subplots()
 
diff --git a/lib/matplotlib/tests/test_style.py b/lib/matplotlib/tests/test_style.py
index aea09462b197..5ed115abb68f 100644
--- a/lib/matplotlib/tests/test_style.py
+++ b/lib/matplotlib/tests/test_style.py
@@ -1,9 +1,9 @@
 from contextlib import contextmanager
-import gc
 from pathlib import Path
 from tempfile import TemporaryDirectory
 import sys
 
+import numpy as np
 import pytest
 
 import matplotlib as mpl
@@ -165,7 +165,7 @@ def test_xkcd_no_cm():
     assert mpl.rcParams["path.sketch"] is None
     plt.xkcd()
     assert mpl.rcParams["path.sketch"] == (1, 100, 2)
-    gc.collect()
+    np.testing.break_cycles()
     assert mpl.rcParams["path.sketch"] == (1, 100, 2)
 
 
diff --git a/src/_c_internal_utils.c b/src/_c_internal_utils.c
index 532824c5d05b..7209b4f396fc 100644
--- a/src/_c_internal_utils.c
+++ b/src/_c_internal_utils.c
@@ -68,7 +68,13 @@ mpl_GetCurrentProcessExplicitAppUserModelID(PyObject* module)
     wchar_t* appid = NULL;
     HRESULT hr = GetCurrentProcessExplicitAppUserModelID(&appid);
     if (FAILED(hr)) {
+#if defined(PYPY_VERSION_NUM) && PYPY_VERSION_NUM < 0x07030600
+        /* Remove when we require PyPy 7.3.6 */
+        PyErr_SetFromWindowsErr(hr);
+        return NULL;
+#else
         return PyErr_SetFromWindowsErr(hr);
+#endif
     }
     PyObject* py_appid = PyUnicode_FromWideChar(appid, -1);
     CoTaskMemFree(appid);
@@ -89,7 +95,13 @@ mpl_SetCurrentProcessExplicitAppUserModelID(PyObject* module, PyObject* arg)
     HRESULT hr = SetCurrentProcessExplicitAppUserModelID(appid);
     PyMem_Free(appid);
     if (FAILED(hr)) {
+#if defined(PYPY_VERSION_NUM) && PYPY_VERSION_NUM < 0x07030600
+        /* Remove when we require PyPy 7.3.6 */
+        PyErr_SetFromWindowsErr(hr);
+        return NULL;
+#else
         return PyErr_SetFromWindowsErr(hr);
+#endif
     }
     Py_RETURN_NONE;
 #else