Tensor categories are natural generalizations of Hopf algebras and they provide a very convenient language for formulating and solving problems both in representation theory and in quantum topology. For example, topological invariants of three-dimensional manifolds can be constructed out of a tensor category with an extra structure. Here, examples are provided by quantum groups at roots of unity. I am interested in the problem of deformation of such categories and will talk about new results in this direction. As it is often in algebra, infinitesimal deformations are controlled by Hochschild type complexes. I will show how to use such complexes in a rather explicit study of deformations of tensor categories and tensor functors arising in Hopf algebra theory.