Recently there has been an increase of interest in the electrodynamics of moving media. This is primarily due to the emergence of new experimental possibilities. High-current accelerators give dense beams of relativistic electrons or of plasma cluster. In addition, moderating structures exist in which the velocity of propagation of electromagnetic waves is much less than the velocity of light in a vacuum, and this facilitates observation of relativistic effects. These successes have rendered realistic the consideration of a whole number of applications. These include the amplification of electromagnetic waves and the augmentation of frequency by reflection from interfaces with moving media, and also the diagnostics of material flows. Various astrophysical and ionosphere applications are of interest. In this review, the fundamental equations of the electrodynamics of moving media are presented. The treatment is carried out by the Mandel'shtam--Tamm method, which permits introduction of potentials and of the Hertz tensor for the description of the electromagnetic field in a moving medium. The laws of propagation of free electromagnetic waves are considered. A derivation is given of Green's function in the electrodynamics of moving media. This makes possible a unified determination of the fields in the presence of sources. Consideration is given to peculiar effects that occur at faster-than-light velocity of motion of the medium (when the velocity of motion of the medium exceeds the phase velocity of light in the rest system of the medium). In the appendix, expressions are found for the fields of a stationary point charge and magnetic dipole, and the Lienard--Wiechert potentials in a moving medium are also obtained.