| In this talk, after motivating the study of singularities in the context of the Minimal Model Program, we will explain the connection between vanishing on Fano-type varieties and cohomological properties of singularities. In particular, we will explain how to construct del Pezzo surfaces in low characteristic violating Kodaira vanishing and how to deduce the existence of klt not Cohen-Macaulay threefold singularities. Time permitting, we will discuss some possible obstructions to the existence of flips. |