  We show that the Kodaira vanishing theorem can fail on smooth Fano varieties
of any characteristic p > 0. Taking cones over some of these varieties, we give
the first examples of terminal singularities which are not Cohen-Macaulay. By a
different method, we construct a terminal singularity of dimension 3 (the
lowest possible) in characteristic 2 which is not Cohen-Macaulay.
