The Maxwell equations, the Minkowski material equations, and the conditions which must be obeyed by fields on moving interfaces are used to obtain expressions for the frequencies and wave vectors of all the waves that appear on reflection and refraction by moving interfaces, by a tangential discontinuity of velocities of two media, and by a moving mirror. The main reported investigations of these topics are discussed. Calculations are made of a large frequency shift, of a considerable increase in the amplitudes, and of a change in the direction of propagation of waves reflected by a rapidly moving mirror or by the front of a parameter traveling in a medium at rest. Allowance is made for the Fresnel transmission of waves across such an interface and for the finite thickness of the transition layer. The change in the parameters of the reflected wave packets is considered together with the problem of the exchange of energy and momentum between an electromagnetic field and a moving mirror. In the case of a tangential discontinuity the rotation of the plane of polarization of reflected and refracted waves is dealt with, as well as the amplification of these waves on reflection from a medium moving faster than light.