After a brief introduction to the geometric features of black holes, I will discuss recent joint work with Dafermos, Rodnianski and Taylor introducing a new scheme to prove small data global existence results for quasilinear wave equations on black hole backgrounds (see arXiv:2212.14093). A key ingredient is a new physical space estimate at the top order which avoids understanding the detailed structure of trapped geodesics. The relation to the stability problem for black holes will also be discussed.