Observational evidence for the existence of shell sources of radio emission is cited, and it is pointed out that present day theories of non-thermal radio sources associated with the remnants of supernovae are not directly applicable to shell sources. Possible circumstances which could give rise to the formation of such sources are outlined. The parameters, some measurable by radio astronomical techniques, which characterize a shell source are listed. The system is treated as a gaseous envelope which has been ejected by a supernova and expands into the interstellar medium, driving a shock wave ahead of it. The equations describing this physical system are given and simplified and it is shown that between shock front and envelope a shell is formed whose thickness varies with the direction relative to the undisturbed interstellar magnetic field. This shell acts as a strong non-thermal radio source due to the trapped relativistic electron flux which is generated within the supernova remnant and, subject to some simple assumptions, the theory enables all important properties of the source to be calculated from its radio astronomical data. The model is applied to the radio sources 3C392 and IC443 and the results are shown to be consistent with values typical of type II supernovae radio sources.