I will start by motivating the study of the linear wave equation on black holes spacetimes and develop the necessary tools for its analysis, most prominently the vectorfield method (Lecture 1). I will then prove the uniform boundedness of waves on Schwarzschild, and at least state and explain the decay statements which have been proven for Schwarzschild and Kerr (Lectures 2+3). In the last lecture, I will explain our recent proof of the linear stability of the Schwarzschild spacetime under gravitational perturbations (Lecture 4). The latter is joint work with Dafermos and Rodnianski.
