Searching for tensor categories

Tensor categories are a generalization of groups which play the same role in quantum mathematics that groups play in ordinary mathematics.  There are several important sources of tensor categories, namely ordinary groups and Drinfel'd-Jimbo quantum groups.  But it's natural to wonder whether there are more exceptional or exotic tensor categories out there.  I will talk about several ways we search for new tensor categories, some of the interesting examples that we've found, and our attempts to fit these examples into patterns.