Kodaira vanshing states that on a smooth projective complex variety the higher cohomology of an ample line bundle twisted by the canonical bundle vanishes. Most proofs of this theorem rely on transcendental methods (i.e. inputs from Hodge Theory). As originally shown by Raynaud, Kodaira vanishing fails to hold in characteristic p. In this talk, we will give a brief introduction to Kodaira's theorem. We will then discuss the ideas behind Raynaud's counterexample. Speaker(s): Harold Blum (UM)