In this talk we will present some of the modern applications of representation theory in the field of knot invariants and categorifications. This includes brief descriptions of finite-dimensional irreducible representations of double affine Hecke algebra, skew-Howe dualities and their generalizations, as well as representation theory of Lie superalgebras gl(n|m). Also some open problems in representation theory and related combinatorics will be presented, which are of particular importance in topology and categorifications.
