Every tensor category has a Drinfeld centre. In nice enough cases, this is a modular tensor category. Fusion categories (the finitely semisimple rigid tensor categories) classify 3-2-1-0 dimensional topological quantum field theories, and their Drinfeld centres determine entirely the 3-2-1 dimensional part. Computing Drinfeld centres is a major challenge in understanding and classifying fusion categories.
While computing the Drinfeld centre is difficult, we've recently realised that you can often get "most of the way there". From any modular tensor category we can extract the $S$ and $T$ matrices, which have rich representation theoretic, combinatorial, and number theoretic properties --- they constitute modular data. As far as anyone knows, the modular data could actually be a complete invariant, although there is no particularly strong reason to believe this yet. I'll describe work with Terry Gannon and Corey Jones on "the modular data machine", an algorithm which in many cases effectively computes the modular data for the Drinfeld centre of a given unitary fusion category.