Black Holes feature among the most fascinating predictions of Einstein?s theory of general rela-tivity. In the past decades fundamental progress has been made both at the experimental level and at the level of their mathematical understanding, which is the topic of my talk. I will begin by connecting general relativity to the analysis of non-linear partial differential equations and describe the the geometry of the most important black hole solutions. In the main part of the talk I will introduce the stability problem for black holes and focus on recent work with Dafer-mos, Rodnianski and Taylor establishing the non-linear stability of the Schwarzschild family of black holes. Emphasis will be put on explaining the interplay between geometry and analysis.