A general technique for solving Maxwell's equations exactly, based on expansion of the solution in a complete set of vector basis functions has been developed. These vector eigenfunctions are derived from the complete set of separable solutions to the scalar Helmholtz equation in a particular coordinate system and are shown to form a complete set. The method is applicable to a variety of problems including the study of near and far field electromagnetic scattering from particles with arbitrary shapes, plasmon resonances in spherical nanoparticles with spherically concentric 'shells' and the calculation of plasmon resonances in the sphere-plane geometry. An exact method for solving the inhomogenous Maxwell's equation (i.e., in the presence of charges and currents) is also outlined.