In the last decades, an interesting variety of extended models of gravity has been proposed with the goal of capturing cosmological effects such as the accelerated phases of expansion and/or the so-called "dark sector" of our universe. In parallel, the quest for a full-fledged theory of quantum gravity proceeds by investigating the low-energy limit of candidate models. Many of these modified gravity models might leave imprints in the physics of compact objects and with gravitational-wave astronomy we have the unprecedented opportunity to test them against data with improving accuracy.
A popular class of models (scalar-tensor theories) extends the field content of general relativity with an additional scalar field. These theories provide multiple examples where black hole and neutron star physics deviates from general relativity and can be constrained with observations. In this sense, superradiance and spontaneous growth of scalar fields around black holes and neutron stars are potentially detectable signatures of new physics. Screening mechanisms can in principle hide scalar effects, but their effectiveness in the strong-field regime is still largely unmodeled. In this thesis I briefly review the traditional tests of gravity, from the weak-field observations to gravitational-wave tests, before moving to discuss in details a collection of personal contributions in modeling the aforementioned scalar effects
