Abstract: In this talk, we present both classical results and recent developments in the theory of composition operators on Lebesgue-type function spaces and on Sobolev spaces. The study of composition operators on Sobolev spaces started many years ago and was motivated by the question: what change of variables does preserve a Sobolev space? In the talk, we pay special attention to the problem of finding necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator. We provide a description of such mappings.
