An important problem in general relativity is understanding ``radiation tails'' in the exterior regions of spacetimes containing black holes. The heuristic picture of what these tails should look like goes back to work of R. Price in 1972. These tails are related to stability properties of black hole exteriors, and also to the details of their inner structure, in particular, the generic presence of weak null singularities inside black holes. In this talk, I shall describe a rigorous proof of Price's power-law decay rates for the collapse of a spherically symmetric self-gravitating scalar field. Applications of these ideas to linear and non-linear wave equations on various fixed black hole backgrounds will also be discussed. This constitutes joint work with I. Rodnianski.