The two lectures will be about a basic question in representation theory, namely determining the irreducible modular representations of Chevalley groups (the simplest example being the general linear group over a finite field). After the classification of finite simple groups (starting with the famous Feit-Thompson theorem) we know that close relatives of these groups represent almost all finite simple groups. The first lecture will be an introduction to this fascinating subject, outlining the parallels and differences to the representation theory of compact Lie groups. I will then recall Lusztig’s character formula, and discuss the difficult problem of determining for which primes it is valid. Here some surprising number theory shows up. In the second lecture I will discuss recent progress on this problem (partly joint with Ben Elias, Xuhua He and Simon Riche). Here the burgeoning theory of categorification and representations of affine Lie algebras play an important role.