Raynaud in 1971 showed that the Kodaira vanishing theorem fails in positive characteristic. However, one may still ask: what are the consequences of the failure of Kodaira vanishing? We will present a theorem due to Kollar, which says that non-vanishing of H1(X,L*) for an ample line bundle L has very strong geometric consequences. The main ingredient of the proof is a construction by Ekedahl of a purely inseparable cover of X with unusual properties. We will also mention applications to the classification of del Pezzo surfaces and Fano threefolds due to Ekedahl and Shepherd-Barron. Speaker(s): Takumi Murayama (UM)