I will discuss recent progress on the problem of computing the dimension of a self-similar set or measure in $\mathbb{R}$ in the presence of non-trivial overlaps. It is thought that unless the overlaps are "exact" (an essentially algebraic condition), the dimension achieves the trivial upper bound. I will present a weakened version of this that confirms the conjecture in some special cases. A key ingredient is a theorem in additive combinatorics that describes in a statistical sense the structure of measures whose convolution has roughly the same entropy at small scales as the original measure. As time permits, I will also discuss the situation in $\mathbb{R}^d$.