The one that's most often talked about is the Principality of Sealand. It's on an old British observation post in the sea. There was a period of time where it was outside of UK territorial waters (I'm sure the link will tell you why), and it was purchased and claimed its own soverignty.
There are a lot of places like this: some are new like Sealand, but most are old city-states that are just accepted. Luxembourg is a good example of