The moving sources are represented in terms of a vector and a scalar potential, in a reference frame in which the sources are at rest. Subsequently, Green's functions pertaining to the Fourier spectra of the radiated fields are found. The spectra of the fields detected by a stationary observer are given as space integrals over the volume occupied by the sources, of the product of these Green's functions with the source current density. The method is also applied to the spectra of the fields scattered by a two-dimensional moving body. In that case the scatterer is replaced by equivalent induced currents and/or charges which are the sources of the scattered field. For the scattering case, the theory is not developed in its full generality but the method is illustrated by the canonical case of a moving perfectly conducting circular cylinder immersed in an incident plane E wave.