The central role black holes are believed to play in our universe is contingent on their stability properties. The stability problem for black holes is a domain that has witnessed rapid mathematical advances over the last 20 years, synthesising older insights from the physics literature with more recent techniques from the geometric theory of hyperbolic pde’s. The problem has both linear and nonlinear aspects and is governed by the interplay between geometrical features of the black hole spacetimes (horizon redshift, unstable photon orbits, etc.), analytic techniques capturing the dispersive properties of waves and the often mysterious algebraic structural properties of the Einstein equations. This talk will overview some of the essential lessons we have learned about the stability of black holes, starting from the Schwarzschild case all the way to the full subextremal (i.e., |a|<M) range of the Kerr family, and some of the things left to do.