Question

I am very far from an expert on the subject, but I have had to answer this question before. I will provide a few of my answers that seem to be omitted from the above discussion.

1. Fusion rings, or fusion rules, appear in certain physical "thought experiments" (though I am not aware that they have yet actually been observed). For instance "anyons" are defined by certain fusion rules,and if they existed would be useful in quantum computation. In the dogma of field theory, for these fusion rules to be physically meaningful, there needs to be a categorification of them, which would be a fusion category. This initiates a broad class of problems: given a fusion ring, decide whether or not it has a categorification, and if so, how many (up to equivalence). By Ocneanu's rigidity theorem, there are only finitely many categorifications of any given fusion ring, so it's possibly a "tame" problem to solve.

2. There is no hope (at present, per Greg's comment =]) to completely classify fusion categories in any sense, so far as I understand. The classification of fusion categories would encompass not only the classification of finite groups, but also of compact Lie groups (via the tilting module construction on the associated quantum group which yields a fusion category). So people classify fusion categories in small classes under the assumption that group theoretical categories (ones defined purely in terms of group theory: representations of groups, cohomology of groups, morita equivalences, etc.) is "easy" and they want to study the difference between the two contexts.

3. Greg mentioned that studying fusion categories is like studying semi-simple Hopf algebras, except that (a) there isn't necessarily a fiber functor to vector spaces, and (b) even if there does abstractly exist one, you don't choose one. If one admits the interest in studying Hopf algebras, then one has to admit the interest in fusion categories as a sort of "basis free" version. A direct application to finite groups is pinning down the precise relation between the groups D_8 and the quaternions. They are obviously not isomorphic; however their group rings are morita equivalent as rings (since they have the same number of irreducibles). Their irreps even have the same dimensions, so one can ask if their fusion categories are equivalent as fusion categories (they are not in this case, but there are some non-isomorphic groups which are so-called "isocategorical" meaning that not only are their group rings isomorphic, but the Hopf algebras are twist equivalent as Hopf algebras. The most sensible way to prove this sort of statement is through fusion categories.

4. For me, I am a fairly concrete-minded person, but someone who nevertheless tries to understand modern algebraic geometry, algebraic topology and category theory as best I can. Fusion categories have been a fantastic discovery for me, because they are in many ways homotopy theoretic/higher category-type constructions, but they are about as simple as one can get (because you basically have constrained the 1-morphisms as much as possible by the semi-simplicity assumption, and just focus on the higher morphisms). So for instance the first 2-groupoid I was ever able to understand in completely concrete terms arises in a paper of Etingof Nikshych and Ostrik about fusion categories. As such they can be viewed as a kindergarten of higher categories.

5. By the way, there is also some interest in "finite tensor categories" which are not semi-simple but satisfy the other finiteness conditions of fusion categories. (so you have finitely many simple objects, and you posit that every object is a finite-length extension of the simples). There's actually a great deal of the theory from fusion categories which generalizes here. So far as I can tell, the only obstacle in developing this notion more completely is that no one has had time to do it yet.