Kodaira and Kawamata-Viehweg vanishing is frequently used to lift sections of adjoint bundles, a crucial part of many arguments in the classification theory of algebraic varieties, notably in many proofs of the Minimal Model Program. These vanishing theorems fail in mixed characteristic situations, for example for 1.) proper, flat schemes over the p-adic numbers, or 2.) proper birational models of mixed characteristic local rings. I present a work that remedies this situation to some extent. In particular, we are able to a.) show Kodaira and Kawamata-Viehweg vanishing in many situations, b.) prove the 3-dimensional Minimal Model Program for excellent schemes, and c.) draw geometric corollaries of point b.) to the existence of the moduli space of stable surfaces in mixed characteristic. This is a joint work with Bhargav Bhatt, Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek.