Given a collection of contracting similarities f_1,...,f_k on R^d, there is a unique compact set K that is equal to the union of f_j(K) for j=1,...,k. We call such a set self-similar. If in addition, a probability vector p_1,...,p_k is given, then there is a unique probability measure mu on R^d such that mu=p_1f_1(mu)+...+p_kf_k(mu). Such a measure is called self-similar. These object are of great interest in fractal geometry and the most fundamental problem about them is to determine their dimension. I will review some recent progress in this area mostly focusing on the case of Bernoulli convolutions. These are self-similar measures on R with respect to the two similarities x->lambda x +1 and x->lambda x - 1, where lambda is a fixed parameter in (0,1).