diff --git a/tests/test_api.py b/tests/test_api.py
index e0b57114b2..07bb9408a7 100755
--- a/tests/test_api.py
+++ b/tests/test_api.py
@@ -181,6 +181,11 @@ def test_sign_verify(self):
         with self.assertRaises(exceptions.UnsignedMetadataError):
             snapshot_key.verify_signature(metadata_obj)
 
+        # Test verifying with explicitly set serializer
+        targets_key.verify_signature(metadata_obj, CanonicalJSONSerializer())
+        with self.assertRaises(exceptions.UnsignedMetadataError):
+            targets_key.verify_signature(metadata_obj, JSONSerializer())
+
         sslib_signer = SSlibSigner(self.keystore['snapshot'])
         # Append a new signature with the unrelated key and assert that ...
         metadata_obj.sign(sslib_signer, append=True)
@@ -200,6 +205,32 @@ def test_sign_verify(self):
         with self.assertRaises(exceptions.UnsignedMetadataError):
             targets_key.verify_signature(metadata_obj)
 
+        # Test failure on unknown scheme (securesystemslib UnsupportedAlgorithmError)
+        scheme = timestamp_key.scheme
+        timestamp_key.scheme = "foo"
+        with self.assertRaises(exceptions.UnsignedMetadataError):
+            timestamp_key.verify_signature(metadata_obj)
+        timestamp_key.scheme = scheme
+
+        # Test failure on broken public key data (securesystemslib CryptoError)
+        public = timestamp_key.keyval["public"]
+        timestamp_key.keyval["public"] = "ffff"
+        with self.assertRaises(exceptions.UnsignedMetadataError):
+            timestamp_key.verify_signature(metadata_obj)
+        timestamp_key.keyval["public"] = public
+
+        # Test failure with invalid signature (securesystemslib FormatError)
+        sig = metadata_obj.signatures[timestamp_keyid]
+        correct_sig = sig.signature
+        sig.signature = "foo"
+        with self.assertRaises(exceptions.UnsignedMetadataError):
+            timestamp_key.verify_signature(metadata_obj)
+
+        # Test failure with valid but incorrect signature
+        sig.signature = "ff"*64
+        with self.assertRaises(exceptions.UnsignedMetadataError):
+            timestamp_key.verify_signature(metadata_obj)
+        sig.signature = correct_sig
 
     def test_metadata_base(self):
         # Use of Snapshot is arbitrary, we're just testing the base class features
diff --git a/tuf/api/metadata.py b/tuf/api/metadata.py
index 73f221a04e..dc30b2465d 100644
--- a/tuf/api/metadata.py
+++ b/tuf/api/metadata.py
@@ -33,6 +33,7 @@
     Union,
 )
 
+from securesystemslib import exceptions as sslib_exceptions
 from securesystemslib import hash as sslib_hash
 from securesystemslib import keys as sslib_keys
 from securesystemslib.signer import Signature, Signer
@@ -483,8 +484,6 @@ def verify_signature(
         Raises:
             UnsignedMetadataError: The signature could not be verified for a
                 variety of possible reasons: see error message.
-            TODO: Various other errors currently bleed through from lower
-                level components: Issue #1351
         """
         try:
             signature = metadata.signatures[self.keyid]
@@ -500,15 +499,25 @@ def verify_signature(
 
             signed_serializer = CanonicalJSONSerializer()
 
-        if not sslib_keys.verify_signature(
-            self.to_securesystemslib_key(),
-            signature.to_dict(),
-            signed_serializer.serialize(metadata.signed),
-        ):
+        try:
+            if not sslib_keys.verify_signature(
+                self.to_securesystemslib_key(),
+                signature.to_dict(),
+                signed_serializer.serialize(metadata.signed),
+            ):
+                raise exceptions.UnsignedMetadataError(
+                    f"Failed to verify {self.keyid} signature",
+                    metadata.signed,
+                )
+        except (
+            sslib_exceptions.CryptoError,
+            sslib_exceptions.FormatError,
+            sslib_exceptions.UnsupportedAlgorithmError,
+        ) as e:
             raise exceptions.UnsignedMetadataError(
-                f"Failed to verify {self.keyid} signature for metadata",
+                f"Failed to verify {self.keyid} signature",
                 metadata.signed,
-            )
+            ) from e
 
 
 class Role: