A modular functor is a system of mapping class group representations on finite-dimensional vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion naturally appears 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 their relation with once-extended topological field theories in dimension three. Afterwards, I will explain a new approach to modular functors via cyclic and modular operads and their bicategorical algebras (up to coherent homotopy). 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. We also obtain a uniqueness result for extensions from genus zero surfaces to all surfaces that generalizes a result of Andersen-Ueno. (The talk is based on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)