diff --git a/asdf/schema.py b/asdf/schema.py
index 2fe3a51a8..ffe5432a8 100644
--- a/asdf/schema.py
+++ b/asdf/schema.py
@@ -79,7 +79,8 @@ def _type_to_tag(type_):
 def validate_tag(validator, tag_pattern, instance, schema):
     """
     Implements the tag validation directive, which checks the
-    tag against a pattern which may include '*' wildcards.
+    tag against a pattern that may include wildcards.  See
+    `asdf.util.uri_match` for details on the matching behavior.
     """
     if hasattr(instance, '_tag'):
         instance_tag = instance._tag
diff --git a/asdf/tests/test_util.py b/asdf/tests/test_util.py
index 83164c94d..5616f2370 100644
--- a/asdf/tests/test_util.py
+++ b/asdf/tests/test_util.py
@@ -49,9 +49,16 @@ def test_patched_urllib_parse():
     ("asdf://somewhere.org/tags/foo-1.0", "asdf://somewhere.org/tags/bar-1.0", False),
     ("asdf://somewhere.org/tags/foo-*", "asdf://somewhere.org/tags/foo-1.0", True),
     ("asdf://somewhere.org/tags/foo-*", "asdf://somewhere.org/tags/bar-1.0", False),
+    ("asdf://somewhere.org/tags/foo-*", "asdf://somewhere.org/tags/foo-extras/bar-1.0", False),
     ("asdf://*/tags/foo-*", "asdf://anywhere.org/tags/foo-4.9", True),
     ("asdf://*/tags/foo-*", "asdf://anywhere.org/tags/bar-4.9", False),
+    ("asdf://*/tags/foo-*", "asdf://somewhere.org/tags/foo-extras/bar-4.9", False),
+    ("asdf://**/*-1.0", "asdf://somewhere.org/tags/foo-1.0", True),
+    ("asdf://**/*-1.0", "asdf://somewhere.org/tags/foo-2.0", False),
+    ("asdf://**/*-1.0", "asdf://somewhere.org/tags/foo-extras/bar-1.0", True),
+    ("asdf://**/*-1.0", "asdf://somewhere.org/tags/foo-extras/bar-2.0", False),
     ("asdf://somewhere.org/tags/foo-*", None, False),
+    ("**", None, False),
 ])
 def test_uri_match(pattern, uri, result):
     assert util.uri_match(pattern, uri) is result
diff --git a/asdf/util.py b/asdf/util.py
index 926b61b40..bb09f760d 100644
--- a/asdf/util.py
+++ b/asdf/util.py
@@ -28,7 +28,7 @@
 
 __all__ = ['human_list', 'get_array_base', 'get_base_uri', 'filepath_to_url',
            'iter_subclasses', 'calculate_padding', 'resolve_name', 'NotSet',
-           'is_primitive']
+           'is_primitive', 'uri_match']
 
 
 def human_list(l, separator="and"):
@@ -456,12 +456,16 @@ def is_primitive(value):
 def uri_match(pattern, uri):
     """
     Determine if a URI matches a URI pattern with possible
-    wildcards.
+    wildcards.  The two recognized wildcards:
+
+    "*":  match any character except /
+
+    "**": match any character
 
     Parameters
     ----------
     pattern : str
-        URI pattern with * wildcards.
+        URI pattern.
     uri : str
         URI to check against the pattern.
 
@@ -474,7 +478,7 @@ def uri_match(pattern, uri):
         return False
 
     if "*" in pattern:
-        return _compile_uri_match_pattern(pattern).match(uri) is not None
+        return _compile_uri_match_pattern(pattern).fullmatch(uri) is not None
     else:
         return pattern == uri
 
@@ -483,5 +487,8 @@ def uri_match(pattern, uri):
 def _compile_uri_match_pattern(pattern):
     # Escape the pattern in case it contains regex special characters
     # ('.' in particular is common in URIs) and then replace the
-    # escaped asterisk with a .* regex matcher.
-    return re.compile(re.escape(pattern).replace(r"\*", ".*"))
+    # escaped asterisks with the appropriate regex matchers.
+    pattern = re.escape(pattern)
+    pattern = pattern.replace(r"\*\*", r".*")
+    pattern = pattern.replace(r"\*", r"[^/]*")
+    return re.compile(pattern)