Knowing whether optimal maps are smooth or not is an important step towards a qualitative understanding of them. In the 90's Caffarelli developed a regularity theory on R^n for the quadratic cost, which was then extended by Ma-Trudinger-Wang and Loeper to general cost functions which satisfy a suitable structural condition. Unfortunately, this condition is very restrictive, and when considered on Riemannian manifolds with the cost given by the squared distance, it is satisfied only in very particular cases. Hence the need to develop a partial regularity theory: is it true that optimal maps are always smooth outside a "small" singular set? The aim of this talk is to first review the "classical" regularity theory for optimal maps, and then describe some recent results about their partial regularity.