Projections of self-similar and self-affine fractals

A classical theorem of Marstrand says that for any Borel set E in the plane of Hausdorff dimension d, the orthogonal projection of E onto a line with slope t has Hausdorff dimension min(d,1), for almost every slope t. In general, one can't say anything more than this - the set of exceptional directions may have full dimension. However, if E has some additional structure, for example if E is a self-similar set, one can hope to determine the set of exceptions precisely. I am going to present some recent progress in this direction, and describe some related problems which remain open. Parts of my talk will be based on joint work with A. Ferguson, T. Jordan and Y. Peres.