In the last years, the possibility of constructing a quantum computer has gotten a lot of attention. There are different models and perspectives about it, one of them has a topological nature which has the advantage of being protected against decoherence.  In this talk, we will introduce the notion of a modular tensor category, which is a rich mathematical structure that models 2-dimensional topological phases of matter, which are a route to topological quantum computation. To get a better understanding of these categories, we will present some important examples and properties and some results related to classification and constructions of modular tensor categories. If time allows, we will also discuss some new directions and open problems in the area.