Modular functors are systems of mapping class group representations. The notion plays an important role in the representation theory of quantum groups and conformal field theory and is closely related to three-dimensional topological field theory. After summarizing the classical constructions, I will outline an approach to modular functors using cyclic and modular operads, as well as factorization homology, that leads to a classification of modular functors. The idea is to state the main result relatively quickly and then to move on to applications (order of Dehn twists in the representations) and interesting special cases (Drinfeld centers). This is based on joint work with Adrien Brochier and Lukas Müller.