  Within Einstein's theory of gravity, any compact object heavier than a few
solar masses must be a black hole. Any observation showing otherwise would
imply either new physics beyond General Relativity or new exotic matter fields
beyond the Standard Model, and might provide a portal to understand some
puzzling properties of a black hole. We give a short overview on tests of the
nature of dark compact objects with present and future gravitational-wave
observations, including inspiral tests of the multipolar structure of compact
objects and of their tidal deformability, ringdown tests, and searches for
near-horizon structures with gravitational-wave echoes.
