In 1990, Saito gave a strong generalization of Kodaira's vanishing theorem using his theory of mixed Hodge modules. In the first half, I want to explain the statement in a special where it can be made reasonably explicit, and also give an easy consequence. In the second half, I will describe a proof, different from Saito's, using characteristic p methods. Speaker(s): Donu Arapura (Purdue University)