A modular functor is defined as a system of mapping class group representations on vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion plays an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will give an introduction to the theory of modular functors and recall some classical constructions. Afterwards, I will explain the approach to modular functors via cyclic and modular operads and their bicategorical algebras. This will allow us to extend the known constructions of modular functors and to classify modular functors by certain cyclic algebras over the little disk operad for which an obstruction formulated in terms of factorization homology vanishes. (The talk is based to a different extent on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)
