Quantum representations of mapping class groups are a class of finite-dimensional representations of mapping class groups that has its origin in quantum algebra (e.g. the representation theory of Hopf algebras) and that often has strong ties to three-dimensional topological field theory. After explaining the interest in these representations from the perspectives of algebra, topology and mathematical physics, I will also give an idea of the construction procedure. Finally, I will explain how systems of mapping class group representations can be formalized using the notion of a modular functor. This leads to my current research projects (joint with Adrien Brochier and Lukas Müller) concerned with an approach to quantum representations of mapping class groups using cyclic and modular operads as well as factorization homology. This approach leads to a classification of modular functors.