Modular tensor categories arise naturally in many areas of mathematics, such as conformal field theory, quantum groups and Hopf algebras, low dimensional topology, representations of braid groups, and they have important applications in condensed matter physics, modeling topological phases of matter. In this talk, I will start by introducing the relevant concepts (modular, braided and fusion categories) together with examples to help us understand better the principal characters of this story. I will also mention some interesting properties of these categories and an overview of the classification of “small” modular tensor categories. At the end, I will present some results about odd-dimensional modular tensor categories based on a joint work with Agustina Czenky.

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