One very general definition of "homotopy theory" is using strict things to get information about weaker ones; for example, using 2-categorical tools to prove bicategorical theorems. The passage from categories of strict objects to categories of weak ones is fairly well understood, but the same process applied to functors still hides some mysteries. We'll start by reviewing several notions of "derived functor" and how to describe them abstractly, and then discuss how to generalize them to an enriched situation. The traditional context is Quillen's theory of model categories, but in fact considerably less structure suffices. Finally, we'll apply these ideas to construct, and abstractly characterize, weighted homotopy limits.