Vanishing theorems for cohomology groups are one of the essential tools of modern algebraic geometry, and have particularly important applications in higher dimensional geometry. Under strong positivity assumptions on line bundles, for example ampleness, there are well-known "standard" vanishing theorems, like those of Kodaira, Nakano and Kawamata-Viehweg. They have very useful partial analogues, called Generic Vanishing Theorems - first discovered by Green and Lazarsfeld - when the positivity hypotheses are weakened. I will describe all of the above and their importance, and then explain that recent techniques based on Fourier-Mukai functors and homological algebra can be used to widely extend the context of generic vanishing, and relate it to standard vanishing. As an application, I will explain how to generalize the results of Green-Lazarsfeld to a version of Kodaira vanishing under weak positivity hypotheses.