Consistent systems of representations of mapping class groups, also called modular functors, play an important role in quantum algebra where they are typically constructed from algebraic data such as a quantum group or a vertex operator algebra. Such representations are sometimes called quantum representations of mapping class groups. In my talk, I will provide an introduction to modular functors, expand on their relation to three-dimensional topological field theories and discuss their construction, their properties and their classification. This talk is based on different joint projects with Adrien Brochier, Lukas Müller, Christoph Schweigert and Yang Yang.