The propagation of electromagnetic waves excited by electric dipoles oriented along the axis of multilayered spheroidal structures of finite conductivity is investigated. The electromagnetic parameters and the thickness of the layers of the structure are assumed to be functions of the latitude. In the analysis, electric and magnetic field transforms that constitute a discrete and a continuous spectrum of spherical waves are used to provide a suitable basis for the expansion of the electromagnetic fields at any point in the irregular spheroidal structure. For spheroidal structures with good conducting cores, the terms in the solutions associated with the continuous part of the wave spectrum vanish. In general, however, when the skin depth for the core is large compared to its dimensions or when the sources are located in the core of the structure and propagation in the core is of special interest, the contribution from the continuous part of the wave spectrum cannot be neglected. At each interface between the layers of the irregular spheroidal structure, exact boundary conditions are imposed. Since the terms of the field expansions in the irregular structure do not individually satisfy the boundary conditions, Maxwell's equations are reduced to sets of coupled ordinary first-order differential equations for the wave amplitudes. The solutions are shown to satisfy the reciprocity relationships in electromagnetic theory. The analysis may be applied to problems of radio wave propagation in a nonuniform model of the earth-ionosphere waveguide, particularly when focusing effects at the antipodes are important.