Studying Lie superalgebras and supergroups was initially motivated by applications in physics. In the recent years interesting connections with other branches of mathematics were discovered. The goal of the talk is to review some of these results. I start with describing four series of algebraic supergroups, which are natural generalizations of general linear, orthogonal and symplectic groups. We shall see different superanalogues of Schur-- Weyl duality, which reveal connections with universal tensor categories constructed by Deligne. Then we discuss geometric methods in representation theory of algebraic supergroups: associated variety and Borel-Weil-Bott theory. Finally, I will talk about categorification and weight diagram technique and try to explain how they can be used for calculating the characters of irreducible representations of classical supergroups.