Tidal Love numbers describe the linear response of compact objects to external tidal perturbations, and they are found to vanish exactly for black holes within General Relativity. We discuss the robustness of this property when non-linearities in the theory are taken into account, and investigate the scenario of black holes immersed in an external environment, addressing the detectability of tidal effects at future gravitational wave experiments. Finally, we discuss the tidal deformability of more exotic compact objects, showing the relevance of universality relations in distinguishing among them.