Groups can be studied through their representations as symmetries of vector spaces. The collection of representations of a given group forms a monoidal category. Fusion categories are monoidal categories which share many properties with the representation categories of finite groups. In addition to finite groups, interesting examples of fusion categories come from quantum groups and from operator algebras.
In this talk we will introduce fusion categories and describe some of their properties. Then we will discuss some exotic examples which appeared in the theory of von Neumann algebras.