Dark matter, black holes, and gravitational waves

The formation and growth of black holes can strongly influence the distribution of dark matter around them. I discuss here the different types of dark matter overdensities around black holes, including dark matter cusps, spikes, mounds, crests, and gravitational atoms. I then review recent results on the evolution of a black holes binary in presence of dark matter, focusing on the energy transfer between binary and dark matter induced by dynamical friction. Finally, I present the prospects for studying dark matter with gravitational wave observations, and argue that future interferometers might be able to detect and characterise dark matter overdensities around black holes.