Although at first sight it may seem an odd idea, I will argue that it is actually quite natural to investigate the properties of General Relativity and its black holes in the limit in which the number of spacetime dimensions grows to infinity. The theory simplifies dramatically: it reduces to a theory of non-interacting particles, of finite radius but vanishingly small cross sections, which do not emit nor absorb radiation of any finite frequency. This leads to efficient calculational approaches in an expansion around this limit, as well as to intriguing connections to low-dimensional string-theory black holes.