My 5 month old waves her hands back and forth in front of her face and likes to sucks on her fist. Why?

It is probably like what the rest have mentioned, your little one is experimenting and being fascinated by her hands. For babies, they learn about different parts of their bodies as they develop. Waving, clenching and releasing, trying to grab things etc are just part of the process of their motor skills development. Sucking on her fist is simply just a self-comforting behavior. You need not be worried about this. :)

My baby is fascinated with her hands and fingers at 6 months. She often stops to admire her fingers and can be entertained for quite a long time! It's part of normal development where they are starting to understand their bodies and movement. Sucking is a very natural instinct for babies, that's why they often use their fist or thumb in place of a nipple.

Sounds like baby is having fun discovering the use of her hands and mouth. Her hands are like toys to her as she discovers that they are actually part of her. Sucking on her fists could be a comforting mechanism or simply, your baby's way of discovering different sensations (her tiny fingers have many sensitive nerve endings).