In the first part of this talk, I'll explain a geometric categorification of the Hecke algebra in terms of perverse sheaves on the flag variety. In the second part, we'll study the affine Hecke algebra. In this case, there are two categorical realizations: one using perverse sheaves (on the affine flag variety) and another using coherent sheaves (on the Langlands dual Steinberg variety). As time permits, I'll explain some properties of these categories, relationships between them, and some applications to representation theory.