The potential functions in optimal transport are, in a suitably weak sense, solutions of a Monge–Ampère type equation. In this talk we discuss the regularity for these equations. We pay particular attention to a necessary condition for regularity, known as A3w, and its geometric interpretation. Then we discuss a more general class of Monge–Ampère type equations which arise from problems in geometric optics. We explain how the highly successful regularity theory for optimal transport has been extended to this new class of equations.