An efficient, exact solution for the complete field structure inside, in, and outside a spherical shell of arbitrary linear media, excited by a monochromatic plane wave is given. The fields in each region are expressed as vector multipole expansions whose coefficients are evaluated through the boundary conditions. The numerical reduction involves the use of modified spherical Bessel functions and recursion relations. The technique is illustrated by numerical results for the fields inside an imperfectly conducting shell. This analysis virtually exhausts the class of problems of spherical shells constructed of linear media.