Physical aspects of the theory of black holes in an external electromagnetic field are reviewed. The ``magnetized'' black hole model is currently widely discussed in astrophysics because it provides a basis for the explanation of the high energy activity of galactic cores and quasars. The particular feature of this model is that it predicts unusual ``gravimagnetic'' phenomena that arise as a result of a natural combination of effects in electrodynamics and gravitation, namely, the appearance of an inductive potential difference during the rotation of a black hole in a magnetic field, the drift of a black hole in an external electromagnetic field, the change in the chemical potential of the event horizon, the creation of an effective ergosphere of a black hole in a magnetic field, and so on. Questions relating to the description of electromagnetic fields in Kerr space-time are examined, including their influence on the space-time metric, the interaction between a rotating charged black hole and an external electromagnetic field, the motion of charged particles near ``magnetized'' black holes, including their spontaneous and stimulated emission, and the influence of magnetic fields on quantum-mechanical processes in black holes.