diff --git a/moto/sns/exceptions.py b/moto/sns/exceptions.py
--- a/moto/sns/exceptions.py
+++ b/moto/sns/exceptions.py
@@ -1,7 +1,13 @@
 from moto.core.exceptions import RESTError
 
 
-class SNSNotFoundError(RESTError):
+class SNSException(RESTError):
+    def __init__(self, *args, **kwargs):
+        kwargs["template"] = "wrapped_single_error"
+        super().__init__(*args, **kwargs)
+
+
+class SNSNotFoundError(SNSException):
     code = 404
 
     def __init__(self, message, **kwargs):
@@ -13,42 +19,42 @@ def __init__(self):
         super().__init__(message="Topic does not exist")
 
 
-class ResourceNotFoundError(RESTError):
+class ResourceNotFoundError(SNSException):
     code = 404
 
     def __init__(self):
         super().__init__("ResourceNotFound", "Resource does not exist")
 
 
-class DuplicateSnsEndpointError(RESTError):
+class DuplicateSnsEndpointError(SNSException):
     code = 400
 
     def __init__(self, message):
         super().__init__("DuplicateEndpoint", message)
 
 
-class SnsEndpointDisabled(RESTError):
+class SnsEndpointDisabled(SNSException):
     code = 400
 
     def __init__(self, message):
         super().__init__("EndpointDisabled", message)
 
 
-class SNSInvalidParameter(RESTError):
+class SNSInvalidParameter(SNSException):
     code = 400
 
     def __init__(self, message):
         super().__init__("InvalidParameter", message)
 
 
-class InvalidParameterValue(RESTError):
+class InvalidParameterValue(SNSException):
     code = 400
 
     def __init__(self, message):
         super().__init__("InvalidParameterValue", message)
 
 
-class TagLimitExceededError(RESTError):
+class TagLimitExceededError(SNSException):
     code = 400
 
     def __init__(self):
@@ -58,14 +64,14 @@ def __init__(self):
         )
 
 
-class InternalError(RESTError):
+class InternalError(SNSException):
     code = 500
 
     def __init__(self, message):
         super().__init__("InternalFailure", message)
 
 
-class TooManyEntriesInBatchRequest(RESTError):
+class TooManyEntriesInBatchRequest(SNSException):
     code = 400
 
     def __init__(self):
@@ -75,7 +81,7 @@ def __init__(self):
         )
 
 
-class BatchEntryIdsNotDistinct(RESTError):
+class BatchEntryIdsNotDistinct(SNSException):
     code = 400
 
     def __init__(self):
diff --git a/moto/sns/models.py b/moto/sns/models.py
--- a/moto/sns/models.py
+++ b/moto/sns/models.py
@@ -51,6 +51,7 @@ def __init__(self, name, sns_backend):
         self.subscriptions_pending = 0
         self.subscriptions_confimed = 0
         self.subscriptions_deleted = 0
+        self.sent_notifications = []
 
         self._policy_json = self._create_default_topic_policy(
             sns_backend.region_name, self.account_id, name
@@ -70,6 +71,9 @@ def publish(self, message, subject=None, message_attributes=None, group_id=None)
                 message_attributes=message_attributes,
                 group_id=group_id,
             )
+        self.sent_notifications.append(
+            (message_id, message, subject, message_attributes, group_id)
+        )
         return message_id
 
     @classmethod
@@ -89,7 +93,7 @@ def physical_resource_id(self):
 
     @property
     def policy(self):
-        return json.dumps(self._policy_json)
+        return json.dumps(self._policy_json, separators=(",", ":"))
 
     @policy.setter
     def policy(self, policy):
@@ -215,7 +219,7 @@ def publish(
                     if value["Type"].startswith("Binary"):
                         attr_type = "binary_value"
                     elif value["Type"].startswith("Number"):
-                        type_value = "{0:g}".format(value["Value"])
+                        type_value = str(value["Value"])
 
                     raw_message_attributes[key] = {
                         "data_type": value["Type"],
@@ -404,6 +408,19 @@ def publish(self, message):
 
 
 class SNSBackend(BaseBackend):
+    """
+    Responsible for mocking calls to SNS. Integration with SQS/HTTP/etc is supported.
+
+    Messages published to a topic are persisted in the backend. If you need to verify that a message was published successfully, you can use the internal API to check the message was published successfully:
+
+    .. sourcecode:: python
+
+        from moto.sns import sns_backend
+        all_send_notifications = sns_backend.topics[topic_arn].sent_notifications
+
+    Note that, as this is an internal API, the exact format may differ per versions.
+    """
+
     def __init__(self, region_name):
         super().__init__()
         self.topics = OrderedDict()
@@ -880,7 +897,7 @@ def publish_batch(self, topic_arn, publish_batch_request_entries):
                     message=entry["Message"],
                     arn=topic_arn,
                     subject=entry.get("Subject"),
-                    message_attributes=entry.get("MessageAttributes", []),
+                    message_attributes=entry.get("MessageAttributes", {}),
                     group_id=entry.get("MessageGroupId"),
                 )
                 successful.append({"MessageId": message_id, "Id": entry["Id"]})
diff --git a/moto/sns/responses.py b/moto/sns/responses.py
--- a/moto/sns/responses.py
+++ b/moto/sns/responses.py
@@ -31,10 +31,13 @@ def _get_tags(self):
         tags = self._get_list_prefix("Tags.member")
         return {tag["key"]: tag["value"] for tag in tags}
 
-    def _parse_message_attributes(self, prefix="", value_namespace="Value."):
+    def _parse_message_attributes(self):
         message_attributes = self._get_object_map(
             "MessageAttributes.entry", name="Name", value="Value"
         )
+        return self._transform_message_attributes(message_attributes)
+
+    def _transform_message_attributes(self, message_attributes):
         # SNS converts some key names before forwarding messages
         # DataType -> Type, StringValue -> Value, BinaryValue -> Value
         transformed_message_attributes = {}
@@ -63,15 +66,18 @@ def _parse_message_attributes(self, prefix="", value_namespace="Value."):
             if "StringValue" in value:
                 if data_type == "Number":
                     try:
-                        transform_value = float(value["StringValue"])
+                        transform_value = int(value["StringValue"])
                     except ValueError:
-                        raise InvalidParameterValue(
-                            "An error occurred (ParameterValueInvalid) "
-                            "when calling the Publish operation: "
-                            "Could not cast message attribute '{0}' value to number.".format(
-                                name
+                        try:
+                            transform_value = float(value["StringValue"])
+                        except ValueError:
+                            raise InvalidParameterValue(
+                                "An error occurred (ParameterValueInvalid) "
+                                "when calling the Publish operation: "
+                                "Could not cast message attribute '{0}' value to number.".format(
+                                    name
+                                )
                             )
-                        )
                 else:
                     transform_value = value["StringValue"]
             elif "BinaryValue" in value:
@@ -382,6 +388,16 @@ def publish_batch(self):
         publish_batch_request_entries = self._get_multi_param(
             "PublishBatchRequestEntries.member"
         )
+        for entry in publish_batch_request_entries:
+            if "MessageAttributes" in entry:
+                # Convert into the same format as the regular publish-method
+                # FROM: [{'Name': 'a', 'Value': {'DataType': 'String', 'StringValue': 'v'}}]
+                # TO  : {'name': {'DataType': 'Number', 'StringValue': '123'}}
+                msg_attrs = {y["Name"]: y["Value"] for y in entry["MessageAttributes"]}
+                # Use the same validation/processing as the regular publish-method
+                entry["MessageAttributes"] = self._transform_message_attributes(
+                    msg_attrs
+                )
         successful, failed = self.backend.publish_batch(
             topic_arn=topic_arn,
             publish_batch_request_entries=publish_batch_request_entries,

diff --git a/tests/terraform-tests.success.txt b/tests/terraform-tests.success.txt
--- a/tests/terraform-tests.success.txt
+++ b/tests/terraform-tests.success.txt
@@ -84,6 +84,7 @@ TestAccAWSProvider
 TestAccAWSRedshiftServiceAccount
 TestAccAWSRolePolicyAttachment
 TestAccAWSSNSSMSPreferences
+TestAccAWSSNSTopicPolicy
 TestAccAWSSageMakerPrebuiltECRImage
 TestAccAWSServiceDiscovery
 TestAccAWSSQSQueuePolicy
diff --git a/tests/test_sns/test_publish_batch.py b/tests/test_sns/test_publish_batch.py
--- a/tests/test_sns/test_publish_batch.py
+++ b/tests/test_sns/test_publish_batch.py
@@ -145,10 +145,43 @@ def test_publish_batch_to_sqs():
     messages.should.contain({"Message": "1"})
     messages.should.contain({"Message": "2", "Subject": "subj2"})
     messages.should.contain(
+        {"Message": "3", "MessageAttributes": {"a": {"Type": "String", "Value": "v"}}}
+    )
+
+
+@mock_sqs
+@mock_sns
+def test_publish_batch_to_sqs_raw():
+    client = boto3.client("sns", region_name="us-east-1")
+    topic_arn = client.create_topic(Name="standard_topic")["TopicArn"]
+    sqs = boto3.resource("sqs", region_name="us-east-1")
+    queue = sqs.create_queue(QueueName="test-queue")
+
+    queue_url = "arn:aws:sqs:us-east-1:{}:test-queue".format(ACCOUNT_ID)
+    client.subscribe(
+        TopicArn=topic_arn,
+        Protocol="sqs",
+        Endpoint=queue_url,
+        Attributes={"RawMessageDelivery": "true"},
+    )
+
+    entries = [
+        {"Id": "1", "Message": "foo",},
         {
-            "Message": "3",
-            "MessageAttributes": [
-                {"Name": "a", "Value": {"DataType": "String", "StringValue": "v"}}
-            ],
-        }
+            "Id": "2",
+            "Message": "bar",
+            "MessageAttributes": {"a": {"DataType": "String", "StringValue": "v"}},
+        },
+    ]
+    resp = client.publish_batch(TopicArn=topic_arn, PublishBatchRequestEntries=entries,)
+
+    resp.should.have.key("Successful").length_of(2)
+
+    received = queue.receive_messages(
+        MaxNumberOfMessages=10, MessageAttributeNames=["All"],
     )
+
+    messages = [(message.body, message.message_attributes) for message in received]
+
+    messages.should.contain(("foo", None))
+    messages.should.contain(("bar", {"a": {"StringValue": "v", "DataType": "String"}}))
diff --git a/tests/test_sns/test_publishing_boto3.py b/tests/test_sns/test_publishing_boto3.py
--- a/tests/test_sns/test_publishing_boto3.py
+++ b/tests/test_sns/test_publishing_boto3.py
@@ -238,6 +238,49 @@ def test_publish_to_sqs_msg_attr_number_type():
     message.body.should.equal("test message")
 
 
+@mock_sqs
+@mock_sns
+def test_publish_to_sqs_msg_attr_different_formats():
+    """
+    Verify different Number-formats are processed correctly
+    """
+    sns = boto3.resource("sns", region_name="us-east-1")
+    topic = sns.create_topic(Name="test-topic")
+    sqs = boto3.resource("sqs", region_name="us-east-1")
+    sqs_client = boto3.client("sqs", region_name="us-east-1")
+    queue_raw = sqs.create_queue(QueueName="test-queue-raw")
+
+    topic.subscribe(
+        Protocol="sqs",
+        Endpoint=queue_raw.attributes["QueueArn"],
+        Attributes={"RawMessageDelivery": "true"},
+    )
+
+    topic.publish(
+        Message="test message",
+        MessageAttributes={
+            "integer": {"DataType": "Number", "StringValue": "123"},
+            "float": {"DataType": "Number", "StringValue": "12.34"},
+            "big-integer": {"DataType": "Number", "StringValue": "123456789"},
+            "big-float": {"DataType": "Number", "StringValue": "123456.789"},
+        },
+    )
+
+    messages_resp = sqs_client.receive_message(
+        QueueUrl=queue_raw.url, MessageAttributeNames=["All"]
+    )
+    message = messages_resp["Messages"][0]
+    message_attributes = message["MessageAttributes"]
+    message_attributes.should.equal(
+        {
+            "integer": {"DataType": "Number", "StringValue": "123"},
+            "float": {"DataType": "Number", "StringValue": "12.34"},
+            "big-integer": {"DataType": "Number", "StringValue": "123456789"},
+            "big-float": {"DataType": "Number", "StringValue": "123456.789"},
+        }
+    )
+
+
 @mock_sns
 def test_publish_sms():
     client = boto3.client("sns", region_name="us-east-1")
@@ -374,9 +417,14 @@ def callback(request):
         TopicArn=topic_arn, Protocol="http", Endpoint="http://example.com/foobar"
     )
 
-    response = conn.publish(
-        TopicArn=topic_arn, Message="my message", Subject="my subject"
-    )
+    conn.publish(TopicArn=topic_arn, Message="my message", Subject="my subject")
+
+    if not settings.TEST_SERVER_MODE:
+        sns_backend.topics[topic_arn].sent_notifications.should.have.length_of(1)
+        notification = sns_backend.topics[topic_arn].sent_notifications[0]
+        _, msg, subject, _, _ = notification
+        msg.should.equal("my message")
+        subject.should.equal("my subject")
 
 
 @mock_sqs
diff --git a/tests/test_sns/test_topics_boto3.py b/tests/test_sns/test_topics_boto3.py
--- a/tests/test_sns/test_topics_boto3.py
+++ b/tests/test_sns/test_topics_boto3.py
@@ -214,7 +214,7 @@ def test_topic_attributes():
     )
 
     attributes = conn.get_topic_attributes(TopicArn=topic_arn)["Attributes"]
-    attributes["Policy"].should.equal('{"foo": "bar"}')
+    attributes["Policy"].should.equal('{"foo":"bar"}')
     attributes["DisplayName"].should.equal("My display name")
     attributes["DeliveryPolicy"].should.equal(
         '{"http": {"defaultHealthyRetryPolicy": {"numRetries": 5}}}'

SNS publish_batch issue with RawMessageDelivery
To reproduce: https://github.com/spulec/moto/pull/4818

The behavior between `sns.publish` and `sns.publish_batch` seems to be slightly different, in `sns.publish` the message attributes are handled with `SNSResponse._parse_message_attributes`.

This leads to an exception being raised in `SNSModel.publish_batch` that's not a `InvalidParameterValue` so the message is not listed in neither `"Failed"` or `"Succeeded"` in the response:
https://github.com/spulec/moto/blob/0a1bb6bae165b38cdec5e299a7f859f2cb27de93/moto/sns/models.py#L887-L888

In `SNSResponse.publish_batch` I tried hardcoding the `"MessageAttributes"` to the same format that's provided by `_parse_message_attributes` which seemed to make it work (besides the attribute being hardcoded of course)


tests/test_sns/test_publishing_boto3.py::test_publish_to_http tests/test_sns/test_topics_boto3.py::test_topic_attributes
tests/test_sns/test_publish_batch.py::test_publish_batch_non_unique_ids tests/test_sns/test_publish_batch.py::test_publish_batch_unknown_topic tests/test_sns/test_topics_boto3.py::test_create_topic_with_attributes tests/test_sns/test_topics_boto3.py::test_create_topic_should_be_of_certain_length tests/test_sns/test_topics_boto3.py::test_topic_fifo_get_attributes tests/test_sns/test_topics_boto3.py::test_topic_get_attributes tests/test_sns/test_topics_boto3.py::test_topic_corresponds_to_region tests/test_sns/test_topics_boto3.py::test_topic_kms_master_key_id_attribute tests/test_sns/test_topics_boto3.py::test_create_topic_with_tags tests/test_sns/test_topics_boto3.py::test_list_tags_for_resource_error tests/test_sns/test_topics_boto3.py::test_create_and_delete_topic tests/test_sns/test_topics_boto3.py::test_create_fifo_topic tests/test_sns/test_topics_boto3.py::test_add_remove_permissions tests/test_sns/test_topics_boto3.py::test_remove_permission_errors tests/test_sns/test_topics_boto3.py::test_create_topic_should_be_indempodent tests/test_sns/test_topics_boto3.py::test_create_topic_must_meet_constraints tests/test_sns/test_publish_batch.py::test_publish_batch_standard_with_message_group_id tests/test_sns/test_publishing_boto3.py::test_publish_subject tests/test_sns/test_publishing_boto3.py::test_publish_bad_sms tests/test_sns/test_topics_boto3.py::test_get_missing_topic tests/test_sns/test_topics_boto3.py::test_topic_paging tests/test_sns/test_topics_boto3.py::test_tag_topic tests/test_sns/test_publish_batch.py::test_publish_batch_too_many_items tests/test_sns/test_topics_boto3.py::test_tag_resource_errors tests/test_sns/test_publishing_boto3.py::test_publish_sms tests/test_sns/test_topics_boto3.py::test_add_permission_errors tests/test_sns/test_topics_boto3.py::test_create_topic_in_multiple_regions tests/test_sns/test_publishing_boto3.py::test_publish_fifo_needs_group_id tests/test_sns/test_publish_batch.py::test_publish_batch_fifo_without_message_group_id tests/test_sns/test_topics_boto3.py::test_untag_topic tests/test_sns/test_topics_boto3.py::test_topic_get_attributes_with_fifo_false tests/test_sns/test_publishing_boto3.py::test_publish_message_too_long tests/test_sns/test_publishing_boto3.py::test_publish_group_id_to_non_fifo tests/test_sns/test_topics_boto3.py::test_delete_non_existent_topic tests/test_sns/test_publishing_boto3.py::test_publish_to_sqs_bad tests/test_sns/test_topics_boto3.py::test_untag_resource_error