In optimal transport theory, one wants to understand optimizing phenomena occurring when transporting mass distributions in Economics, Physics, Probability, Analysis, Geometry, and Biology. In this talk, we will discuss continuity of optimal transport maps, in view of a pseudo-Riemannian structure which we have formulated recently. Curvature of this geometry plays an essential role for continuity of optimal transportation. This natural geometric framework provides new methods, elementary proofs and extensions of some key ingredients in the regularity theory of Ma, Trudinger, Wang, and Loeper. It also provides new examples, perspectives and research directions. This is joint work with Robert McCann (University of Toronto).