We present a theoretical model to describe spin transport in a structure consisting of a ferromagnetic metal injector, a thin (usually undoped) semiconductor layer, and a ferromagnetic metal collector. In thermal equilibrium the magnetic contacts are spin-polarized whereas the semiconductor is unpolarized. Due to the large ratio of the metal to semiconductor conductivities, the semiconductor needs to be driven far out of local thermal equilibrium to achieve efficient injection of spin-polarized electrons. This requires a barrier to injection that may be due either to a large Schottky barrier or to an insulating tunnel barrier. Since carrier mobilities (and other relevant parameters) in inorganic and organic semiconductors differ by orders of magnitude, the conditions for achieving a state far from equilibrium at the injecting contact are quite different for the two types of materials.