This paper presents a theory of extragalactic radio sources whose linear dimensions are large compared with their associated galaxies. A catastrophic event in the nucleus of a galaxy is assumed to release a large amount of energy which is converted over a short time scale ( T ∼ 10 5 years) in an unknown way into a relativistic gas with an inverse power law differential energy distribution. This relativistic gas of high energy density expands isotropically into the local extragalactic medium which consists of a tenuous magnetic plasma. The magnetic field of the medium couples the motion of the relativistic gas with that of the thermal gas. As long as the expansion velocity is super-magneto-sonic a shock front separates the undisturbed medium from the expanding region. The essential unknowns of the problem are the medium's hydrogen number density n0 , its magnetic field strength B0 , the total initial energy of the relativistic gas E i , the ratio of the energy in relativistic protons and heavier nuclei to the energy in relativistic electrons and positrons, τ , and a parameter β which is a measure of the instantaneous expansion velocity. The five independent relations required to solve for these unknowns are provided by: the dynamics of the expansion; synchrotron emission theory; depolarization due to differential Faraday rotation; the theory of radio spectral curvature and a stability condition imposed on the external medium. The theory, which requires extensive observational data to be of use, is applied to four well-explored sources. The lack of spherical symmetry in radio galaxy brightness distributions is attributed to an anisotropy of the medium. Provided the undisturbed magnetic field is reasonably well aligned, the theoretical brightness distributions are in agreement with recent interferometric data. In this model, halo and double sources are physically similar but differently oriented with respect to the observer. The polarization properties of these sources are given by the theory. The model accounts adequately both for the lack of unique empirical relations and for the trends which are exhibited by the parameters of a sufficiently large sample of sources. The theory suggests that extended radio galaxies can be formed only in clusters and that to account for the radio properties such clusters require a relatively strong magnetic field (∼ 4 × 10 −6 gauss) and high intergalactic mass densities ( n H ∼ 5 × 10 −4 cm −3 ). Alternative models are considered but found not to account for observed optical and radio features. Some further implications of this theory are examined with reference to observational and theoretical problems in recent literature.