I will motivate why and how General Relativity can be classically extended and how black hole solutions in these extensions can differ from the Schwarzschild and Kerr geometries. I will review how these differences can be perturbative or non-perturbative according to the specific extension of General Relativity; how they can appear already in static configurations or just in time dependent ones (e.g. in binaries); and how they may even challenge the very notion of black hole horizon in specific cases. I will also discuss ways to test these possibilities experimentally, e.g. with gravitational and electromagnetic detectors.