This dissertation considers several aspects of the structure and dynamics of electromagnetic fields around black holes. The four-dimensional, covariant laws of electrodynamics are reformulated in a 3 + 1 (space+time) language in which the key quantities are three-dimensional vectors lying in hypersurfaces of a constant global time t. This formulation is applied to the Blandford-Znajek model of power generation in quasars, which consists of a supermassive black hole surrounded by an accretion disk that holds a magnetic field on the hole, with the rotational energy and angular momentum of the hole and disk being extracted by electromagnetic torques. The 3 + 1 formalism allows the theory of stationary, axisymmetric black holes and their magnetospheres to be couched in an "absolute-space/universal-time" language very similar to the flat spacetime theory of pulsar electrodynamics; and this similarity allows fiat-space pulsar concepts to be extended to curved-space black holes. The Blandford-Znajek quasar model is reformulated in terms of a DC circuit-theory analysis, and action principles describing the overall structure of the magnetosphere and the field distribution on the horizon are developed. A general prescription for constructing global models of force-free magnetospheres is developed and this prescription is used to generate numerical models ofblack-hole magnetospheres for a variety of field configurationsand black-holeangular velocities. The electromagnetic boundary conditions at the horizon of a black hole are described in terms of a recently developed "membrane viewpoint". The necessity and efficacy of using a "stretched horizon" in the membrane viewpoint is discussed, and is illustrated by two simple dynamical problems involving electromagnetic fields near black-hole horizons.
