Light bosonic degrees of freedom have become a serious candidate for dark matter, which seems to pervade our entire universe. The evolution of these fields around curved spacetimes is poorly understood but is expected to display interesting effects. In particular, the interaction of light bosonic fields with supermassive black holes, key players in most galaxies, could provide colourful examples of superradiance and nonlinear bosenova-like collapse. In turn, the observation of spinning black holes is expected to impose stringent bounds on the mass of putative massive bosonic fields in our universe.Our purpose here is to present a comprehensive study of the evolution of linearized massive scalar and vector fields in the vicinities of rotating black holes. The evolution of generic initial data has a very rich structure, depending on the mass of the field and of the black hole. Quasi-normal ringdown or exponential decay followed by a power-law tail at very late times is a generic feature of massless fields at intermediate times. Massive fields generically show a transition to power-law tails early on. For a certain boson field mass range, the field can become trapped in a potential barrier outside the horizon and transition to a bound state. Because there are a number of such quasi-bound states, the generic outcome is an amplitude modulated sinusoidal, or beating, signal, whose envelope is well described by the two lowest overtones. We believe that the appearance of such beatings has gone unnoticed in the past, and in fact mistaken for exponential growth. The amplitude modulation of the signal depends strongly on the relative excitation of the overtones, which in turn is strongly tied to the bound-state geography.A fine tuning of the initial data allows one to see the evolution of the nearly pure bound state mode which turns unstable for sufficiently large black hole rotation. For the first time we explore massive vector fields in generic BH background which are hard, if not impossible, to separate in the Kerr background. Our results show that spinning BHs are generically strongly unstable against massive vector fields.