This paper presents a theory of electromagnetic energy extraction from rotating Kerr black holes. When a rotating black hole is threaded by magnetic field lines supported by external currents in an equatorial disc, an electric potential difference is induced. If the field strength is sufficient, the vacuum becomes unstable, leading to a cascade of electron-positron pair production and the formation of a force-free magnetosphere. In this scenario, energy and angular momentum are extracted electromagnetically. The paper shows that charge does not significantly contribute to the geometry of a rotating black hole. The fundamental equations governing a stationary axisymmetric magnetosphere around a Kerr black hole are derived. The paper discusses the energy and angular momentum balance, and develops a perturbation technique for slowly rotating holes. Solutions are described for field lines threading the hole on conical and paraboloidal surfaces at large distances. The theory is applied to a model of active galactic nuclei containing a massive black hole surrounded by a magnetized accretion disc. In this model, relativistic electrons can be accelerated at large distances from the hole, avoiding significant losses. If the field lines have paraboloidal shape, energy is beamed along antiparallel directions, consistent with observations of compact and extended radio sources. The paper discusses the pair production discharge mechanism, showing that the vacuum can break down due to the creation of sufficient electron-positron pairs. The necessary field strengths for this mechanism are derived, and it is shown that inverse Compton scattering of ambient radiation can prevent electrons from achieving sufficient energies for pair creation. However, within this environment, a more efficient breakdown mechanism exists. The paper derives the fundamental equations governing a force-free magnetosphere in Kerr spacetime. These equations are used to show that energy and angular momentum can be extracted from a rotating black hole. The electromagnetic angular velocity is shown to be constant and equal to half the hole's angular velocity. The paper also discusses the energy and angular momentum transport, showing that energy flows outwards along the poloidal field surfaces. The paper presents a perturbation method for calculating approximate solutions for slowly rotating holes. Two such solutions are derived, and the application of these ideas to a model of active galactic nuclei is outlined. The paper concludes that the efficiency of energy extraction is determined by the electromagnetic angular velocity, which is constant and equal to half the hole's angular velocity.This paper presents a theory of electromagnetic energy extraction from rotating Kerr black holes. When a rotating black hole is threaded by magnetic field lines supported by external currents in an equatorial disc, an electric potential difference is induced. If the field strength is sufficient, the vacuum becomes unstable, leading to a cascade of electron-positron pair production and the formation of a force-free magnetosphere. In this scenario, energy and angular momentum are extracted electromagnetically. The paper shows that charge does not significantly contribute to the geometry of a rotating black hole. The fundamental equations governing a stationary axisymmetric magnetosphere around a Kerr black hole are derived. The paper discusses the energy and angular momentum balance, and develops a perturbation technique for slowly rotating holes. Solutions are described for field lines threading the hole on conical and paraboloidal surfaces at large distances. The theory is applied to a model of active galactic nuclei containing a massive black hole surrounded by a magnetized accretion disc. In this model, relativistic electrons can be accelerated at large distances from the hole, avoiding significant losses. If the field lines have paraboloidal shape, energy is beamed along antiparallel directions, consistent with observations of compact and extended radio sources. The paper discusses the pair production discharge mechanism, showing that the vacuum can break down due to the creation of sufficient electron-positron pairs. The necessary field strengths for this mechanism are derived, and it is shown that inverse Compton scattering of ambient radiation can prevent electrons from achieving sufficient energies for pair creation. However, within this environment, a more efficient breakdown mechanism exists. The paper derives the fundamental equations governing a force-free magnetosphere in Kerr spacetime. These equations are used to show that energy and angular momentum can be extracted from a rotating black hole. The electromagnetic angular velocity is shown to be constant and equal to half the hole's angular velocity. The paper also discusses the energy and angular momentum transport, showing that energy flows outwards along the poloidal field surfaces. The paper presents a perturbation method for calculating approximate solutions for slowly rotating holes. Two such solutions are derived, and the application of these ideas to a model of active galactic nuclei is outlined. The paper concludes that the efficiency of energy extraction is determined by the electromagnetic angular velocity, which is constant and equal to half the hole's angular velocity.