diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index c236f4e2b..d32628338 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -7,8 +7,8 @@ jobs:
     strategy:
       matrix:
         os: [ubuntu-20.04, windows-2019]
-        python-version: ["3.12", "3.11", "3.10", "3.9", "3.8", "3.7", "3.6"]
-        environment: ['3.8', '3.12', '3.11', '3.10', '3.9', '3.7', '3.6', 'interpreter']
+        python-version: ["3.13", "3.12", "3.11", "3.10", "3.9", "3.8", "3.7", "3.6"]
+        environment: ['3.8', '3.13', '3.12', '3.11', '3.10', '3.9', '3.7', '3.6', 'interpreter']
     steps:
       - name: Checkout code
         uses: actions/checkout@v4
diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index fca944293..cf6810fa0 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -6,6 +6,8 @@ Changelog
 Unreleased
 ++++++++++
 
+- Python 3.13 support
+
 0.19.1 (2023-10-02)
 +++++++++++++++++++
 
diff --git a/jedi/_compatibility.py b/jedi/_compatibility.py
index 13a74b7b3..48563d00d 100644
--- a/jedi/_compatibility.py
+++ b/jedi/_compatibility.py
@@ -5,11 +5,24 @@
 import errno
 import sys
 import pickle
+from typing import Any
+
+
+class Unpickler(pickle.Unpickler):
+    def find_class(self, module: str, name: str) -> Any:
+        # Python 3.13 moved pathlib implementation out of __init__.py as part of
+        # generalising its implementation. Ensure that we support loading
+        # pickles from 3.13 on older version of Python. Since 3.13 maintained a
+        # compatible API, pickles from older Python work natively on the newer
+        # version.
+        if module == 'pathlib._local':
+            module = 'pathlib'
+        return super().find_class(module, name)
 
 
 def pickle_load(file):
     try:
-        return pickle.load(file)
+        return Unpickler(file).load()
     # Python on Windows don't throw EOF errors for pipes. So reraise them with
     # the correct type, which is caught upwards.
     except OSError:
diff --git a/jedi/api/environment.py b/jedi/api/environment.py
index 278899aaa..64b318e1f 100644
--- a/jedi/api/environment.py
+++ b/jedi/api/environment.py
@@ -22,7 +22,7 @@
 
 _VersionInfo = namedtuple('VersionInfo', 'major minor micro')  # type: ignore[name-match]
 
-_SUPPORTED_PYTHONS = ['3.12', '3.11', '3.10', '3.9', '3.8', '3.7', '3.6']
+_SUPPORTED_PYTHONS = ['3.13', '3.12', '3.11', '3.10', '3.9', '3.8', '3.7', '3.6']
 _SAFE_PATHS = ['/usr/bin', '/usr/local/bin']
 _CONDA_VAR = 'CONDA_PREFIX'
 _CURRENT_VERSION = '%s.%s' % (sys.version_info.major, sys.version_info.minor)
diff --git a/jedi/inference/__init__.py b/jedi/inference/__init__.py
index aadfeba96..bd31cbd39 100644
--- a/jedi/inference/__init__.py
+++ b/jedi/inference/__init__.py
@@ -90,7 +90,7 @@ def __init__(self, project, environment=None, script_path=None):
         self.compiled_subprocess = environment.get_inference_state_subprocess(self)
         self.grammar = environment.get_grammar()
 
-        self.latest_grammar = parso.load_grammar(version='3.12')
+        self.latest_grammar = parso.load_grammar(version='3.13')
         self.memoize_cache = {}  # for memoize decorators
         self.module_cache = imports.ModuleCache()  # does the job of `sys.modules`.
         self.stub_module_cache = {}  # Dict[Tuple[str, ...], Optional[ModuleValue]]
diff --git a/setup.py b/setup.py
index 43292a4be..54b5b0a12 100755
--- a/setup.py
+++ b/setup.py
@@ -35,8 +35,8 @@
       long_description=readme,
       packages=find_packages(exclude=['test', 'test.*']),
       python_requires='>=3.6',
-      # Python 3.11 & 3.12 grammars are added to parso in 0.8.3
-      install_requires=['parso>=0.8.3,<0.9.0'],
+      # Python 3.13 grammars are added to parso in 0.8.4
+      install_requires=['parso>=0.8.4,<0.9.0'],
       extras_require={
           'testing': [
               'pytest<9.0.0',
@@ -101,6 +101,7 @@
           'Programming Language :: Python :: 3.10',
           'Programming Language :: Python :: 3.11',
           'Programming Language :: Python :: 3.12',
+          'Programming Language :: Python :: 3.13',
           'Topic :: Software Development :: Libraries :: Python Modules',
           'Topic :: Text Editors :: Integrated Development Environments (IDE)',
           'Topic :: Utilities',
diff --git a/test/test_api/test_interpreter.py b/test/test_api/test_interpreter.py
index c435db9c4..1aa027bf4 100644
--- a/test/test_api/test_interpreter.py
+++ b/test/test_api/test_interpreter.py
@@ -310,8 +310,9 @@ def test_completion_param_annotations():
     # Need to define this function not directly in Python. Otherwise Jedi is too
     # clever and uses the Python code instead of the signature object.
     code = 'def foo(a: 1, b: str, c: int = 1.0) -> bytes: pass'
-    exec(code, locals())
-    script = jedi.Interpreter('foo', [locals()])
+    exec_locals = {}
+    exec(code, exec_locals)
+    script = jedi.Interpreter('foo', [exec_locals])
     c, = script.complete()
     sig, = c.get_signatures()
     a, b, c = sig.params
@@ -323,7 +324,7 @@ def test_completion_param_annotations():
     assert b.description == 'param b: str'
     assert c.description == 'param c: int=1.0'
 
-    d, = jedi.Interpreter('foo()', [locals()]).infer()
+    d, = jedi.Interpreter('foo()', [exec_locals]).infer()
     assert d.name == 'bytes'
 
 
@@ -525,10 +526,14 @@ def func(a, b, c):
     c = functools.partial(func, 1, c=2)
 
     sig, = jedi.Interpreter(code, [locals()]).get_signatures()
-    assert sig.name == 'partial'
     assert [p.name for p in sig.params] == expected
     assert index == sig.index
 
+    if sys.version_info < (3, 13):
+        # Python 3.13.0b3 makes functools.partial be a descriptor, which breaks
+        # Jedi's `py__name__` detection; see https://github.com/davidhalter/jedi/issues/2012
+        assert sig.name == 'partial'
+
 
 def test_type_var():
     """This was an issue before, see Github #1369"""
diff --git a/test/test_inference/test_signature.py b/test/test_inference/test_signature.py
index f8f71581a..4a1fcb62e 100644
--- a/test/test_inference/test_signature.py
+++ b/test/test_inference/test_signature.py
@@ -1,5 +1,5 @@
 from textwrap import dedent
-from operator import ge, lt
+from operator import eq, ge, lt
 import re
 import os
 
@@ -14,7 +14,8 @@
         ('import math; math.cos', 'cos(x, /)', ['x'], ge, (3, 6)),
 
         ('next', 'next(iterator, default=None, /)', ['iterator', 'default'], lt, (3, 12)),
-        ('next', 'next()', [], ge, (3, 12)),
+        ('next', 'next()', [], eq, (3, 12)),
+        ('next', 'next(iterator, default=None, /)', ['iterator', 'default'], ge, (3, 13)),
 
         ('str', "str(object='', /) -> str", ['object'], ge, (3, 6)),
 
diff --git a/test/test_utils.py b/test/test_utils.py
index acc1df9c2..15a791744 100644
--- a/test/test_utils.py
+++ b/test/test_utils.py
@@ -73,15 +73,21 @@ def test_import(self):
         import os
         s = 'from os import '
         goal = {s + el for el in dir(os)}
+
         # There are minor differences, e.g. the dir doesn't include deleted
         # items as well as items that are not only available on linux.
         difference = set(self.complete(s)).symmetric_difference(goal)
+        ACCEPTED_DIFFERENCE_PREFIXES = [
+            '_', 'O_', 'EX_', 'EFD_', 'MFD_', 'TFD_',
+            'SF_', 'ST_', 'CLD_', 'POSIX_SPAWN_', 'P_',
+            'RWF_', 'CLONE_', 'SCHED_', 'SPLICE_',
+        ]
         difference = {
             x for x in difference
-            if all(not x.startswith('from os import ' + s)
-                   for s in ['_', 'O_', 'EX_', 'MFD_', 'SF_', 'ST_',
-                             'CLD_', 'POSIX_SPAWN_', 'P_', 'RWF_',
-                             'CLONE_', 'SCHED_'])
+            if not any(
+                x.startswith('from os import ' + prefix)
+                for prefix in ACCEPTED_DIFFERENCE_PREFIXES
+            )
         }
         # There are quite a few differences, because both Windows and Linux
         # (posix and nt) libraries are included.