diff --git a/CHANGES.rst b/CHANGES.rst
index a20b4e3e8..5945a2666 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -17,7 +17,7 @@
 - Drop support for Python 3.5. [#856]
 
 - Add new extension API to support versioned extensions.
-  [#850]
+  [#850, #851]
 
 2.7.0 (2020-07-23)
 ------------------
diff --git a/asdf/asdf.py b/asdf/asdf.py
index 3547cc2f4..f2275122a 100644
--- a/asdf/asdf.py
+++ b/asdf/asdf.py
@@ -113,10 +113,12 @@ def __init__(self, tree=None, uri=None, extensions=None, version=None,
             files follow custom conventions beyond those enforced by the
             standard.
         """
+        # Don't use the version setter here; it tries to access
+        # the extensions, which haven't been assigned yet.
         if version is None:
-            self.version = get_config().default_version
+            self._version = versioning.AsdfVersion(get_config().default_version)
         else:
-            self.version = version
+            self._version = versioning.AsdfVersion(validate_version(version))
 
         self.extensions = extensions
 
@@ -190,6 +192,9 @@ def version(self, value):
         value : str or asdf.versioning.AsdfVersion
         """
         self._version = versioning.AsdfVersion(validate_version(value))
+        # Reassigning extensions cues the processing that checks each
+        # extension's ASDF Standard requirement.
+        self.extensions = self.extensions
 
     @property
     def version_string(self):
@@ -354,11 +359,20 @@ def _get_extension(e):
 
         requested_extensions = [_get_extension(e) for e in requested_extensions]
 
+        for extension in requested_extensions:
+            if self.version_string not in extension.asdf_standard_requirement:
+                warnings.warn(
+                    "Extension {} does not support ASDF Standard {}.  It has been disabled.".format(
+                        extension, self.version_string
+                    ),
+                    AsdfWarning
+                )
+
         extensions = []
         # Add requested extensions to the list first, so that they
         # take precedence.
         for extension in requested_extensions + get_config().extensions:
-            if extension not in extensions:
+            if extension not in extensions and self.version_string in extension.asdf_standard_requirement:
                 extensions.append(extension)
 
         return extensions
diff --git a/asdf/extension/_extension.py b/asdf/extension/_extension.py
index 683a1aa48..8ce9e2564 100644
--- a/asdf/extension/_extension.py
+++ b/asdf/extension/_extension.py
@@ -1,3 +1,5 @@
+from packaging.specifiers import SpecifierSet
+
 from ..util import get_class_name
 from ._legacy import AsdfExtension
 
@@ -34,6 +36,14 @@ def __init__(self, delegate, package_name=None, package_version=None):
             else:
                 raise TypeError("Extension property 'legacy_class_names' must contain str values")
 
+        value = getattr(self._delegate, "asdf_standard_requirement", None)
+        if isinstance(value, str):
+            self._asdf_standard_requirement = SpecifierSet(value)
+        elif value is None:
+            self._asdf_standard_requirement = SpecifierSet()
+        else:
+            raise TypeError("Extension property 'asdf_standard_requirement' must be str or None")
+
         if self._legacy:
             self._legacy_class_names.add(self._class_name)
 
@@ -62,6 +72,17 @@ def legacy_class_names(self):
         """
         return self._legacy_class_names
 
+    @property
+    def asdf_standard_requirement(self):
+        """
+        Get the extension's ASDF Standard requirement.
+
+        Returns
+        -------
+        packaging.specifiers.SpecifierSet
+        """
+        return self._asdf_standard_requirement
+
     @property
     def types(self):
         """
diff --git a/asdf/tests/test_asdf.py b/asdf/tests/test_asdf.py
index f366a7b4e..54e96d3a1 100644
--- a/asdf/tests/test_asdf.py
+++ b/asdf/tests/test_asdf.py
@@ -15,12 +15,14 @@ def __init__(
         self,
         extension_uri=None,
         legacy_class_names=None,
+        asdf_standard_requirement=None,
         types=None,
         tag_mapping=None,
         url_mapping=None,
     ):
         self._extension_uri = extension_uri
         self._legacy_class_names = set() if legacy_class_names is None else legacy_class_names
+        self._asdf_standard_requirement = asdf_standard_requirement
         self._types = [] if types is None else types
         self._tag_mapping = [] if tag_mapping is None else tag_mapping
         self._url_mapping = [] if url_mapping is None else url_mapping
@@ -45,6 +47,10 @@ def extension_uri(self):
     def legacy_class_names(self):
         return self._legacy_class_names
 
+    @property
+    def asdf_standard_requirement(self):
+        return self._asdf_standard_requirement
+
 
 def test_asdf_file_version():
     with config_context() as config:
@@ -130,6 +136,43 @@ def test_asdf_file_extensions():
         AsdfFile(extensions="not-a-URI")
 
 
+def test_asdf_file_version_requirement():
+    extension_with_requirement = TestExtension(
+        extension_uri="asdf://somewhere.org/extensions/foo-1.0",
+        asdf_standard_requirement="==1.5.0",
+    )
+
+    # No warnings if the requirement is fulfilled:
+    with assert_no_warnings():
+        AsdfFile(version="1.5.0", extensions=[extension_with_requirement])
+
+    # Version doesn't match the requirement, so we should see a warning
+    # and the extension should not be enabled:
+    with pytest.warns(AsdfWarning, match="does not support ASDF Standard 1.4.0"):
+        af = AsdfFile(version="1.4.0", extensions=[extension_with_requirement])
+        assert ExtensionProxy(extension_with_requirement) not in af.extensions
+
+    # Version initially matches the requirement, but changing
+    # the version on the AsdfFile invalidates it:
+    af = AsdfFile(version="1.5.0", extensions=[extension_with_requirement])
+    assert ExtensionProxy(extension_with_requirement) in af.extensions
+    with pytest.warns(AsdfWarning, match="does not support ASDF Standard 1.4.0"):
+        af.version = "1.4.0"
+    assert ExtensionProxy(extension_with_requirement) not in af.extensions
+
+    # Extension registered with the config should not provoke
+    # a warning:
+    with config_context() as config:
+        config.add_extension(extension_with_requirement)
+        with assert_no_warnings():
+            af = AsdfFile(version="1.4.0")
+            assert ExtensionProxy(extension_with_requirement) not in af.extensions
+
+        # ... unless the user explicitly requested the invalid exception:
+        with pytest.warns(AsdfWarning, match="does not support ASDF Standard 1.4.0"):
+            af = AsdfFile(version="1.4.0", extensions=[extension_with_requirement])
+
+
 def test_open_asdf_extensions(tmpdir):
     extension = TestExtension(extension_uri="asdf://somewhere.org/extensions/foo-1.0")
 
diff --git a/asdf/tests/test_extension.py b/asdf/tests/test_extension.py
index c8ad8e9c0..2c0c0db7c 100644
--- a/asdf/tests/test_extension.py
+++ b/asdf/tests/test_extension.py
@@ -1,4 +1,5 @@
 import pytest
+from packaging.specifiers import SpecifierSet
 
 from asdf.extension import (
     BuiltinExtension,
@@ -9,6 +10,7 @@
 
 from asdf.tests.helpers import assert_extension_correctness
 
+
 def test_builtin_extension():
     extension = BuiltinExtension()
     assert_extension_correctness(extension)
@@ -42,6 +44,7 @@ def test_proxy_legacy():
 
     assert proxy.extension_uri is None
     assert proxy.legacy_class_names == {"asdf.tests.test_extension.LegacyExtension"}
+    assert proxy.asdf_standard_requirement == SpecifierSet()
     assert proxy.types == [LegacyType]
     assert proxy.tag_mapping == LegacyExtension.tag_mapping
     assert proxy.url_mapping == LegacyExtension.url_mapping
diff --git a/setup.cfg b/setup.cfg
index 938cc20a1..acd3d104d 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -27,6 +27,7 @@ install_requires =
     jsonschema>=3.0.2,<4
     numpy>=1.10
     importlib_resources>=3;python_version<"3.9"
+    packaging>=16.0
 
 [options.extras_require]
 all =