The question of regularity in optimal transport and related equations of Monge Ampere type has seen a lot of activity in the past few decades. Starting from the usual quadratic cost in R^n and now ranging arbitrary costs in Riemannian manifolds (and the related reflector antenna problems).  In this talk, we will give an impressionistic description of Caffarelli's regularity theory for the Monge Ampere equation in Euclidean space, which strongly uses the affine invariance of the equation. We will see when and how such a theory can be pushed to general costs,. The new observation is  that in general regularity arises not so much from affine invariance, but rather from two opposite inequalities for the Mahler volume of c-convex sets  (a kind of generalized Blaschke-Santaló inequalities).  The validity of such inequalities are closely tied to the fourth order Ma-Trudinger-Wang tensor of the cost but they do not require the C^4 regularity of the cost. Based on joint work with Jun Kitagawa.