diff --git a/.github/workflows/testing.yml b/.github/workflows/testing.yml
index 6b43e22..3c77f06 100644
--- a/.github/workflows/testing.yml
+++ b/.github/workflows/testing.yml
@@ -11,10 +11,10 @@ jobs:
     runs-on: ubuntu-latest
     strategy:
       matrix:
-        python-version: [3.8, 3.9, "3.10", 3.11, 3.12]
+        python-version: [3.8, 3.9, "3.10", 3.11, 3.12, 3.13]
     steps:
       - uses: actions/checkout@v2
-      
+
       - name: Set up Python ${{ matrix.python-version }}
         uses: actions/setup-python@v2
         with:
diff --git a/s3path/current_version.py b/s3path/current_version.py
index b77f83e..b96a5c6 100644
--- a/s3path/current_version.py
+++ b/s3path/current_version.py
@@ -57,6 +57,10 @@ class PureS3Path(PurePath):
     S3 is not a file-system but we can look at it like a POSIX system.
     """
     _flavour = flavour
+
+    if sys.version_info >= (3, 13):
+        parser = _S3Flavour()
+
     __slots__ = ()
 
     def __init__(self, *args):
@@ -70,7 +74,10 @@ def __init__(self, *args):
                 new_parts.remove(part)
 
         self._raw_paths = new_parts
-        self._load_parts()
+        if sys.version_info >= (3, 13):
+            self._drv, self._root, self._tail_cached = self._parse_path(self._raw_path)
+        else:
+            self._load_parts()
 
     @classmethod
     def from_uri(cls, uri: str):
@@ -276,7 +283,7 @@ def is_mount(self) -> Literal[False]:
         return False
 
 
-class S3Path(_PathNotSupportedMixin, Path, PureS3Path):
+class S3Path(_PathNotSupportedMixin, PureS3Path, Path):
     def stat(self, *, follow_symlinks: bool = True) -> accessor.StatResult:
         """
         Returns information about this path (similarly to boto3's ObjectSummary).
@@ -433,6 +440,27 @@ def iterdir(self): # todo: -> Generator[S3Path, None, None]:
         for name in accessor.listdir(self):
             yield self._make_child_relpath(name)
 
+    def _make_child_relpath(self, name):
+        # _make_child_relpath was removed from Python 3.13 in
+        # 30f0643e36d2c9a5849c76ca0b27b748448d0567
+        if sys.version_info < (3, 13):
+            return super()._make_child_relpath(name)
+
+        path_str = str(self)
+        tail = self._tail
+        if tail:
+            path_str = f'{path_str}{self._flavour.sep}{name}'
+        elif path_str != '.':
+            path_str = f'{path_str}{name}'
+        else:
+            path_str = name
+        path = self.with_segments(path_str)
+        path._str = path_str
+        path._drv = self.drive
+        path._root = self.root
+        path._tail_cached = tail + [name]
+        return path
+
     def open(
             self,
             mode: Literal['r', 'w', 'rb', 'wb'] = 'r',
@@ -452,7 +480,7 @@ def open(
             errors=errors,
             newline=newline)
 
-    def glob(self, pattern: str):  # todo: -> Generator[S3Path, None, None]:
+    def glob(self, pattern: str, *, case_sensitive=None, recurse_symlinks=False):  # todo: -> Generator[S3Path, None, None]:
         """
         Glob the given relative pattern in the Bucket / key prefix represented by this path,
         yielding all matching files (of any kind)
@@ -461,11 +489,11 @@ def glob(self, pattern: str):  # todo: -> Generator[S3Path, None, None]:
         general_options = accessor.configuration_map.get_general_options(self)
         glob_new_algorithm = general_options['glob_new_algorithm']
         if not glob_new_algorithm:
-            yield from super().glob(pattern)
+            yield from self._glob_py313(pattern)
             return
         yield from self._glob(pattern)
 
-    def rglob(self, pattern: str):  # todo: -> Generator[S3Path, None, None]:
+    def rglob(self, pattern: str, *, case_sensitive=None, recurse_symlinks=False):  # todo: -> Generator[S3Path, None, None]:
         """
         This is like calling S3Path.glob with "**/" added in front of the given relative pattern
         """
@@ -670,6 +698,9 @@ class VersionedS3Path(PureVersionedS3Path, S3Path):
     << VersionedS3Path('/<bucket>/<key>', version_id='<version_id>')
     """
 
+    def __init__(self, *args, version_id):
+        super().__init__(*args)
+
 
 def _is_wildcard_pattern(pat):
     # Whether this pattern needs actual matching using fnmatch, or can
diff --git a/tests/test_path_operations.py b/tests/test_path_operations.py
index 5f84b71..ae3d75b 100644
--- a/tests/test_path_operations.py
+++ b/tests/test_path_operations.py
@@ -471,13 +471,13 @@ def test_iterdir(s3_mock):
 
     s3_path = S3Path('/test-bucket/docs')
     assert sorted(s3_path.iterdir()) == [
-        S3Path('/test-bucket/docs/Makefile'),
         S3Path('/test-bucket/docs/_build'),
         S3Path('/test-bucket/docs/_static'),
         S3Path('/test-bucket/docs/_templates'),
         S3Path('/test-bucket/docs/conf.py'),
         S3Path('/test-bucket/docs/index.rst'),
         S3Path('/test-bucket/docs/make.bat'),
+        S3Path('/test-bucket/docs/Makefile'),
     ]