Classical group representation theory deals with group actions on linear spaces; we consider group actions on compact convex spaces, preserving topological and convex structure. We focus on irreducible actions, and show that for a large class of groups - including connected Lie groups - these can be determined. There is a close connection between this and the theory of bounded harmonic functions on symmetric spaces and their boundary values. A remarkable phenomenon is the fact that for SL(2,R), up to isomorphism, there exists a unique non-degenerate irreducible affine representation.