I will discuss the problem of computing the Hausdorff dimension of self-affine sets and measures, that is, the attractors of families of affine contractions and stationary measures on them. This simple model already presents significant challenges. In joint work with Balazs Barany and Ariel Rapaport we recently settled the main case of this in the plane. In this talk, I will describe the statement and a discuss a little about what goes into the proof, which involves the theory of random matrix products, Ledrappier-Young theory and additive combinatorics.