What types of hardware firewall are routed?

Hey, i read many threads about firewalls, i know that there are many types of firewall at almost every layer except for the First layer.

There are 2 layer firewall that are called transparent that filter packets without being a router hop

There also 3 layer firewall that contains router function

I also know that there are 4 and 7 layer firewas, are these types of devices routed? If they aren't how they works?

What are the most common type of hardware firewall? Is It routed?