diff --git a/amqtt/broker.py b/amqtt/broker.py
index 80245ab..066d76b 100644
--- a/amqtt/broker.py
+++ b/amqtt/broker.py
@@ -183,6 +183,8 @@ class Broker:
         self._subscriptions: dict[str, list[tuple[Session, int]]] = {}
         self._retained_messages: dict[str, RetainedApplicationMessage] = {}
 
+        self._topic_filter_matchers: dict[str, re.Pattern[str]] = {}
+
         # Broadcast queue for outgoing messages
         self._broadcast_queue: asyncio.Queue[dict[str, Any]] = asyncio.Queue()
         self._broadcast_task: asyncio.Task[Any] | None = None
@@ -690,6 +692,11 @@ class Broker:
                 f"[MQTT-3.3.2-2] - {client_session.client_id} invalid TOPIC sent in PUBLISH message, closing connection",
             )
             return False
+        if app_message.topic.startswith("$"):
+            self.logger.warning(
+                f"[MQTT-4.7.2-1] - {client_session.client_id} cannot use a topic with a leading $ character."
+            )
+            return False
 
         permitted = await self._topic_filtering(client_session, topic=app_message.topic, action=Action.PUBLISH)
         if not permitted:
@@ -908,9 +915,6 @@ class Broker:
         self.logger.debug(f"Processing broadcast message: {broadcast}")
 
         for k_filter, subscriptions in self._subscriptions.items():
-            if broadcast["topic"].startswith("$") and (k_filter.startswith(("+", "#"))):
-                self.logger.debug("[MQTT-4.7.2-1] - ignoring broadcasting $ topic to subscriptions starting with + or #")
-                continue
 
             # Skip all subscriptions which do not match the topic
             if not self._matches(broadcast["topic"], k_filter):
@@ -1039,11 +1043,21 @@ class Broker:
         )
 
     def _matches(self, topic: str, a_filter: str) -> bool:
+        if topic.startswith("$") and (a_filter.startswith(("+", "#"))):
+            self.logger.debug("[MQTT-4.7.2-1] - ignoring broadcasting $ topic to subscriptions starting with + or #")
+            return False
+
         if "#" not in a_filter and "+" not in a_filter:
             # if filter doesn't contain wildcard, return exact match
             return a_filter == topic
-        # else use regex
-        match_pattern = re.compile(re.escape(a_filter).replace("\\#", "?.*").replace("\\+", "[^/]*").lstrip("?"))
+
+        # else use regex (re.compile is an expensive operation, store the matcher for future use)
+        if a_filter not in self._topic_filter_matchers:
+            self._topic_filter_matchers[a_filter] = re.compile(re.escape(a_filter)
+                                                               .replace("\\#", "?.*")
+                                                               .replace("\\+", "[^/]*")
+                                                               .lstrip("?"))
+        match_pattern = self._topic_filter_matchers[a_filter]
         return bool(match_pattern.fullmatch(topic))
 
     def _get_handler(self, session: Session) -> BrokerProtocolHandler | None:
diff --git a/amqtt/mqtt/protocol/handler.py b/amqtt/mqtt/protocol/handler.py
index eff191c..66bc1cb 100644
--- a/amqtt/mqtt/protocol/handler.py
+++ b/amqtt/mqtt/protocol/handler.py
@@ -520,8 +520,9 @@ class ProtocolHandler(Generic[C]):
                 elif packet.fixed_header.packet_type == DISCONNECT and isinstance(packet, DisconnectPacket):
                     task = asyncio.create_task(self.handle_disconnect(packet))
                 elif packet.fixed_header.packet_type == CONNECT and isinstance(packet, ConnectPacket):
-                    # TODO: why is this not like all other inside create_task?
-                    await self.handle_connect(packet)  # task = asyncio.create_task(self.handle_connect(packet))
+                    # q: why is this not like all other inside a create_task?
+                    # a: the connection needs to be established before any other packet tasks for this new session are scheduled
+                    await self.handle_connect(packet)
                 if task:
                     running_tasks.append(task)
             except MQTTError:
diff --git a/amqtt/mqtt/publish.py b/amqtt/mqtt/publish.py
index 5d368c7..b3972da 100644
--- a/amqtt/mqtt/publish.py
+++ b/amqtt/mqtt/publish.py
@@ -12,7 +12,7 @@ class PublishVariableHeader(MQTTVariableHeader):
 
     def __init__(self, topic_name: str, packet_id: int | None = None) -> None:
         super().__init__()
-        if "*" in topic_name:
+        if "#" in topic_name or "+" in topic_name:
             msg = "[MQTT-3.3.2-2] Topic name in the PUBLISH Packet MUST NOT contain wildcard characters."
             raise MQTTError(msg)
         self.topic_name = topic_name
diff --git a/tests/mqtt/test_connack.py b/tests/mqtt/test_connack.py
new file mode 100644
index 0000000..58de854
--- /dev/null
+++ b/tests/mqtt/test_connack.py
@@ -0,0 +1,25 @@
+import pytest
+from amqtt.errors import AMQTTError
+from amqtt.mqtt.connack import ConnackPacket
+from amqtt.mqtt.packet import MQTTFixedHeader, PUBLISH
+
+
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        _ = ConnackPacket(fixed=header)
+
+
+@pytest.mark.parametrize("prop", [
+    "return_code",
+    "session_parent"
+])
+def test_empty_variable_header(prop):
+    packet = ConnackPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
diff --git a/tests/mqtt/test_connect.py b/tests/mqtt/test_connect.py
index f552d46..4f1385a 100644
--- a/tests/mqtt/test_connect.py
+++ b/tests/mqtt/test_connect.py
@@ -1,9 +1,11 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
 from amqtt.mqtt.connect import ConnectPacket, ConnectPayload, ConnectVariableHeader
-from amqtt.mqtt.packet import CONNECT, MQTTFixedHeader
+from amqtt.mqtt.packet import CONNECT, MQTTFixedHeader, PUBLISH
 
 
 class ConnectPacketTest(unittest.TestCase):
@@ -150,3 +152,36 @@ class ConnectPacketTest(unittest.TestCase):
         assert message.username == "user"
         assert message.payload.password == "password"
         assert message.password == "password"
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        _ = ConnectPacket(fixed=header)
+
+@pytest.mark.parametrize("prop", [
+    "proto_name",
+    "proto_level",
+    "username_flag",
+    "password_flag",
+    "clean_session_flag",
+    "will_retain_flag",
+    "will_qos",
+    "will_flag",
+    "reserved_flag",
+    "client_id",
+    "client_id_is_random",
+    "will_topic",
+    "will_message",
+    "username",
+    "password",
+    "keep_alive",
+])
+def test_empty_variable_header(prop):
+    packet = ConnectPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
+
diff --git a/tests/mqtt/test_puback.py b/tests/mqtt/test_puback.py
index d2f3a17..bce08c7 100644
--- a/tests/mqtt/test_puback.py
+++ b/tests/mqtt/test_puback.py
@@ -1,7 +1,10 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
+from amqtt.mqtt import PUBLISH, MQTTFixedHeader
 from amqtt.mqtt.puback import PacketIdVariableHeader, PubackPacket
 
 
@@ -20,3 +23,22 @@ class PubackPacketTest(unittest.TestCase):
         publish = PubackPacket(variable_header=variable_header)
         out = publish.to_bytes()
         assert out == b"@\x02\x00\n"
+
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        connect_packet = PubackPacket(fixed=header)
+
+
+@pytest.mark.parametrize("prop", [
+    "packet_id",
+])
+def test_empty_variable_header(prop):
+    connect_packet = PubackPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(connect_packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(connect_packet, prop, "a value")
diff --git a/tests/mqtt/test_pubcomp.py b/tests/mqtt/test_pubcomp.py
index 86ea7ca..8f0b69b 100644
--- a/tests/mqtt/test_pubcomp.py
+++ b/tests/mqtt/test_pubcomp.py
@@ -1,7 +1,10 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
+from amqtt.mqtt import MQTTFixedHeader, PUBLISH
 from amqtt.mqtt.pubcomp import PacketIdVariableHeader, PubcompPacket
 
 
@@ -20,3 +23,21 @@ class PubcompPacketTest(unittest.TestCase):
         publish = PubcompPacket(variable_header=variable_header)
         out = publish.to_bytes()
         assert out == b"p\x02\x00\n"
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        _ = PubcompPacket(fixed=header)
+
+
+@pytest.mark.parametrize("prop", [
+    "packet_id"
+])
+def test_empty_variable_header(prop):
+    packet = PubcompPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
diff --git a/tests/mqtt/test_publish.py b/tests/mqtt/test_publish.py
index 71a115d..c67e0c5 100644
--- a/tests/mqtt/test_publish.py
+++ b/tests/mqtt/test_publish.py
@@ -1,7 +1,10 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
+from amqtt.mqtt.packet import MQTTFixedHeader, CONNECT
 from amqtt.mqtt.constants import QOS_0, QOS_1, QOS_2
 from amqtt.mqtt.publish import PublishPacket, PublishPayload, PublishVariableHeader
 
@@ -116,3 +119,28 @@ class PublishPacketTest(unittest.TestCase):
         assert packet.dup_flag
         assert packet.qos == QOS_2
         assert packet.retain_flag
+
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(CONNECT, 0x00)
+    with pytest.raises(AMQTTError):
+        _ = PublishPacket(fixed=header)
+
+def test_set_flags():
+    packet = PublishPacket()
+    packet.set_flags(dup_flag=True, qos=QOS_1, retain_flag=True)
+
+
+@pytest.mark.parametrize("prop", [
+    "packet_id",
+    "data",
+    "topic_name"
+])
+def test_empty_variable_header(prop):
+    packet = PublishPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
diff --git a/tests/mqtt/test_pubrec.py b/tests/mqtt/test_pubrec.py
index 08f8088..0911570 100644
--- a/tests/mqtt/test_pubrec.py
+++ b/tests/mqtt/test_pubrec.py
@@ -1,7 +1,10 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
+from amqtt.mqtt.packet import MQTTFixedHeader, PUBLISH
 from amqtt.mqtt.pubrec import PacketIdVariableHeader, PubrecPacket
 
 
@@ -20,3 +23,22 @@ class PubrecPacketTest(unittest.TestCase):
         publish = PubrecPacket(variable_header=variable_header)
         out = publish.to_bytes()
         assert out == b"P\x02\x00\n"
+
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        _ = PubrecPacket(fixed=header)
+
+
+@pytest.mark.parametrize("prop", [
+    "packet_id"
+])
+def test_empty_variable_header(prop):
+    packet = PubrecPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
diff --git a/tests/mqtt/test_pubrel.py b/tests/mqtt/test_pubrel.py
index 2c89d49..a803540 100644
--- a/tests/mqtt/test_pubrel.py
+++ b/tests/mqtt/test_pubrel.py
@@ -1,7 +1,10 @@
 import asyncio
 import unittest
+import pytest
 
 from amqtt.adapters import BufferReader
+from amqtt.errors import AMQTTError
+from amqtt.mqtt.packet import MQTTFixedHeader, PUBLISH
 from amqtt.mqtt.pubrel import PacketIdVariableHeader, PubrelPacket
 
 
@@ -20,3 +23,22 @@ class PubrelPacketTest(unittest.TestCase):
         publish = PubrelPacket(variable_header=variable_header)
         out = publish.to_bytes()
         assert out == b"b\x02\x00\n"
+
+
+def test_incorrect_fixed_header():
+    header = MQTTFixedHeader(PUBLISH, 0x00)
+    with pytest.raises(AMQTTError):
+        _  = PubrelPacket(fixed=header)
+
+
+@pytest.mark.parametrize("prop", [
+    "packet_id"
+])
+def test_empty_variable_header(prop):
+    packet = PubrelPacket()
+
+    with pytest.raises(ValueError):
+        assert getattr(packet, prop) is not None
+
+    with pytest.raises(ValueError):
+        assert setattr(packet, prop, "a value")
\ No newline at end of file
diff --git a/tests/test_broker.py b/tests/test_broker.py
index 93e3f9f..a73c4d9 100644
--- a/tests/test_broker.py
+++ b/tests/test_broker.py
@@ -510,17 +510,62 @@ async def test_client_publish_dup(broker):
 
 
 @pytest.mark.asyncio
-async def test_client_publish_invalid_topic(broker):
+async def test_client_publishing_invalid_topic(broker):
     assert broker.transitions.is_started()
-    pub_client = MQTTClient()
+    pub_client = MQTTClient(config={'auto_reconnect': False})
     ret = await pub_client.connect("mqtt://127.0.0.1/")
     assert ret == 0
 
-    await pub_client.publish("/+", b"data", QOS_0)
+    await pub_client.subscribe([
+        ("my/+/topic", QOS_0)
+    ])
+    await asyncio.sleep(0.5)
+
+    # need to build & send packet directly to bypass client's check of invalid topic name
+    #    see test_client.py::test_publish_to_incorrect_wildcard for client checks
+    packet = PublishPacket.build(topic_name='my/topic', message=b'messages',
+                                 packet_id=None, dup_flag=False, qos=QOS_0, retain=False)
+    packet.topic_name = "my/+/topic"
+    await pub_client._handler._send_packet(packet)
+    await asyncio.sleep(0.5)
+
+    with pytest.raises(asyncio.TimeoutError):
+        msg = await pub_client.deliver_message(timeout_duration=1)
+        assert msg is None
+
     await asyncio.sleep(0.1)
     await pub_client.disconnect()
 
 
+@pytest.mark.asyncio
+async def test_client_publish_asterisk(broker):
+    """'*' is a valid, non-wildcard character for MQTT."""
+    assert broker.transitions.is_started()
+    pub_client = MQTTClient(config={'auto_reconnect': False})
+    ret = await pub_client.connect("mqtt://127.0.0.1/")
+    assert ret == 0
+
+    await pub_client.subscribe([
+        ("my*/topic", QOS_0),
+        ("my/+/topic", QOS_0)
+    ])
+    await asyncio.sleep(0.1)
+    await pub_client.publish('my*/topic', b'my valid message', QOS_0, retain=False)
+    await asyncio.sleep(0.1)
+    msg = await pub_client.deliver_message(timeout_duration=1)
+    assert msg is not None
+    assert msg.topic == "my*/topic"
+    assert msg.data == b'my valid message'
+
+    await asyncio.sleep(0.1)
+    msg = await pub_client.publish('my/****/topic', b'my valid message', QOS_0, retain=False)
+    assert msg is not None
+    assert msg.topic == "my/****/topic"
+    assert msg.data == b'my valid message'
+
+    await pub_client.disconnect()
+
+
 @pytest.mark.asyncio
 async def test_client_publish_big(broker, mock_plugin_manager):
     pub_client = MQTTClient()
diff --git a/tests/test_client.py b/tests/test_client.py
index 4859c4e..f0975ab 100644
--- a/tests/test_client.py
+++ b/tests/test_client.py
@@ -7,7 +7,7 @@ import pytest
 
 from amqtt.broker import Broker
 from amqtt.client import MQTTClient
-from amqtt.errors import ClientError, ConnectError
+from amqtt.errors import ClientError, ConnectError, MQTTError
 from amqtt.mqtt.constants import QOS_0, QOS_1, QOS_2
 
 formatter = "[%(asctime)s] %(name)s {%(filename)s:%(lineno)d} %(levelname)s - %(message)s"
@@ -490,3 +490,17 @@ async def test_client_no_auth():
             await client.connect("mqtt://127.0.0.1:1883/")
 
         await broker.shutdown()
+
+
+@pytest.mark.asyncio
+async def test_publish_to_incorrect_wildcard(broker_fixture):
+    client = MQTTClient(config={'auto_reconnect': False})
+    await client.connect("mqtt://127.0.0.1/")
+
+    with pytest.raises(MQTTError):
+        await client.publish("my/+/topic", b'plus-sign wildcard topic invalid publish')
+    with pytest.raises(MQTTError):
+        await client.publish("topic/#", b'hash wildcard topic invalid publish')
+
+    await client.publish("topic/*", b'asterisk topic normal publish')
+    await client.disconnect()
diff --git a/tests/test_dollar_topics.py b/tests/test_dollar_topics.py
new file mode 100644
index 0000000..374f548
--- /dev/null
+++ b/tests/test_dollar_topics.py
@@ -0,0 +1,110 @@
+import asyncio
+import logging
+
+import pytest
+
+from amqtt.broker import Broker
+from amqtt.client import MQTTClient
+from amqtt.mqtt.constants import QOS_0
+
+
+logger = logging.getLogger(__name__)
+
+@pytest.mark.asyncio
+async def test_publish_to_dollar_sign_topics():
+    """Applications cannot use a topic with a leading $ character for their own purposes [MQTT-4.7.2-1]."""
+
+    cfg = {
+        'listeners': {'default': {'type': 'tcp', 'bind': '127.0.0.1'}},
+        'plugins': {'amqtt.plugins.authentication.AnonymousAuthPlugin': {"allow_anonymous": True}},
+    }
+
+    b = Broker(config=cfg)
+    await b.start()
+    await asyncio.sleep(0.1)
+    c = MQTTClient(config={'auto_reconnect': False})
+    await c.connect()
+    await asyncio.sleep(0.1)
+    await c.subscribe(
+        [('$#', QOS_0),
+         ('#', QOS_0)]
+    )
+    await asyncio.sleep(0.1)
+    await c.publish('$MY', b'message should be blocked')
+    await asyncio.sleep(0.1)
+
+    with pytest.raises(asyncio.TimeoutError):
+        # wait long enough for broker sys plugin to run
+        _ = await c.deliver_message(timeout_duration=1)
+
+    await c.disconnect()
+    await asyncio.sleep(0.1)
+    await b.shutdown()
+
+@pytest.mark.asyncio
+async def test_hash_will_not_receive_dollar():
+    """A subscription to “#” will not receive any messages published to a topic beginning with a $ [MQTT-4.7.2-1]."""
+
+    cfg = {
+        'listeners': {'default': {'type': 'tcp', 'bind': '127.0.0.1'}},
+        'plugins': {
+            'amqtt.plugins.authentication.AnonymousAuthPlugin': {"allow_anonymous": True},
+            'amqtt.plugins.sys.broker.BrokerSysPlugin': {"sys_interval": 2}
+        }
+    }
+
+    b = Broker(config=cfg)
+    await b.start()
+    await asyncio.sleep(0.1)
+    c = MQTTClient(config={'auto_reconnect': False})
+    await c.connect()
+    await asyncio.sleep(0.1)
+    await c.subscribe(
+        [('#', QOS_0)]
+    )
+    await asyncio.sleep(0.1)
+
+    with pytest.raises(asyncio.TimeoutError):
+        # wait long enough for broker sys plugin to run
+        _ = await c.deliver_message(timeout_duration=5)
+
+    await c.disconnect()
+    await asyncio.sleep(0.1)
+    await b.shutdown()
+
+
+@pytest.mark.asyncio
+async def test_plus_will_not_receive_dollar():
+    """A subscription to “+/monitor/Clients” will not receive any messages published to “$SYS/monitor/Clients [MQTT-4.7.2-1]"""
+    # BrokerSysPlugin doesn't use $SYS/monitor/Clients, so this is an equivalent test with $SYS/broker topics
+
+    cfg = {
+        'listeners': {'default': {'type': 'tcp', 'bind': '127.0.0.1'}},
+        'plugins': {
+            'amqtt.plugins.authentication.AnonymousAuthPlugin': {"allow_anonymous": True},
+            'amqtt.plugins.sys.broker.BrokerSysPlugin': {"sys_interval": 2}
+        }
+    }
+
+    b = Broker(config=cfg)
+    await b.start()
+    await asyncio.sleep(0.1)
+    c = MQTTClient(config={'auto_reconnect': False})
+    await c.connect()
+    await asyncio.sleep(0.1)
+    await c.subscribe(
+        [('+/broker/#', QOS_0),
+         ('+/broker/time', QOS_0),
+         ('+/broker/clients/#', QOS_0),
+         ('+/broker/+/maximum', QOS_0)
+         ]
+    )
+    await asyncio.sleep(0.1)
+
+    with pytest.raises(asyncio.TimeoutError):
+        # wait long enough for broker sys plugin to run
+        _ = await c.deliver_message(timeout_duration=5)
+
+    await c.disconnect()
+    await asyncio.sleep(0.1)
+    await b.shutdown()