The minimal model program aims to classify all varieties up to birational equivalence. It is completely understood for surfaces. Actually, we even have a classification of them. Enriques and Kodaira proved it over the complex numbers, while a bit later Bombieri and Mumford extended their results in positive characteristic. Even though it seems like we do not have many differences between the two situations, at a closer look we find some surprises. The Frobenius gives rise to inseparable morphisms which allow us to do new constructions. At the end of the talk, I will show you how to use it to find counterexamples to Kodaira vanishing theorem in positive characteristics.