The coupling between scalar and vector fields has a long and interesting history. Axions are one key possibility to solve the strong CP problem, and axionlike particles could be one solution to the dark matter puzzle. Extensive experimental and observational efforts are actively looking for "axionic" imprints. Given the nature of the coupling, and the universality of free fall, nontrivial important effects are expected in regions where gravity is strong. Rotating black holes (immersed, or not in magnetic fields) are a prime example of such regions. Here, we show the following: (i) A background electromagnetic field induces an axionic instability in flat space, for electric fields above a certain threshold value. Conversely, a homogeneous harmonic axion field induces an instability in the Maxwell sector. When carried over to curved spacetime, this phenomenon translates into generic instabilities of charged black holes. We describe the instability and its likely final state, new black hole solutions. (ii) In the presence of charge, black hole uniqueness results are lost. We find solutions that are small deformations of the Kerr-Newman geometry and hairy stationary solutions without angular momentum but which are "dragged" by the axion. Axion fields must exist around spinning black holes if these are immersed in external magnetic fields. The axion profile can be obtained perturbatively from the electrovacuum solution derived by Wald. (iii) Ultralight axions trigger superradiant instabilities of spinning black holes and form an axionic cloud in the exterior geometry. The superradiant growth can be interrupted or suppressed through couplings such as E . B (typical axionic coupling) but also more generic terms such as direct couplings to the invariant E-2 - B-2. These couplings lead to periodic bursts of light, which occur throughout the history of energy extraction from the black hole. We provide numerical and simple analytical estimates for the rates of these processes. (iv) Finally, we discuss how plasma effects can affect the evolution of superradiant instabilities.