In this talk I will present a purely variational approach to the regularity theory of optimal transportation introduced by Goldman and Otto. The approach closely follows De Giorgi's strategy for the regularity theory of minimal surfaces: at its core lies a Campanato iteration, which allows one to transfer the scaling law of the local transport energy to small scales. In regularity theory, this typically leads to Schauder estimates; but the same idea can also be used to study the local energy scaling of minimizers of non-convex variational problems related to branching phenomena in strongly uniaxial ferromagnets and type-I superconductors in the intermediate state. I will highlight this connection and give a brief overview of further recent developments and point out some other interesting applications. ______________________________ Invited by Prof. Phan Thành Nam