That would depend entirely upon the situation. If the bully is being physically threatening, then I would take legal action against the bully. We are adults, and thankfully, there are consequences for our actions.
If the bully is saying really hurtful things, that are not in a semi humorous tone, then I might ignore the bully completely, like he/she does not exist. Or I might say something rude back, like "Does it really still make you feel better about yourself to put others down? I thought you might have grown out of it by now!".
Some people don't know how to interact with other people without being rude. If that is the case for your bully, then I would use some playful banter to build rapport. Some people actually LIKE IT when you are rude to them! I don't understand it, but I even have some close friends that are like that. When I try to be sweet, or caring in any way, they get angry and mean. But when I am bantering with them, they look at me with adoration. It's totally bizarre!
It helps to know why someone does the things they do. But you don't have to have a complete understanding of what motivates someone in order to interact with them, or build rapport.
Try to think of interacting with the bully as an experiment. That will take away a lot of the personalization of the situation. You are the scientist, trying to figure out what makes this weird creature tick - what is it that you can do to make this creature behave differently? And then have fun with it! Just make sure that you don't get caught up in the fun of it and turn into a bully yourself...