Modular tensor categories

I'll begin describing what modular tensor categories are, and how they are related to other notions in category theory and condensed matter physics. They are ridiculously overconstrained objects, with representation theory, number theory, and topology pulling them in different directions. Currently there are not many known examples (essentially, quantum groups at roots of unity, Drinfeld centres, and a few mild constructions based on these), but I'm sure there are stranger things out there.