The prospects of the direct detection of gravitational waves (GW)  from compact objects that fall into massive black holes (MBHs) raise key questions that are challenging our understanding of dynamics near a MBH: what are the event rates? what are the orbital properties of the GW sources? The basic dynamical properties of this deceptively simple system - a dominating central mass surrounded by an isotropic cluster of test particles - are still not understood: what processes brings these objects to inspiral orbits? What is the equilibrium state of the cluster? Is it in equilibrium at all? I will give an overview of recent progress in addressing these questions and understanding their wider implications, drawing on theoretical solutions, results from simulations, and observations of stars near the MBH in our Galaxy.