A phenomenological theory is developed for multicomponent diffusion, including thermal diffusion, in gas mixtures in which the components may have different temperatures. The theory is based on the hydrodynamic approach of Maxwell and Stefan, as extended and elaborated by Furry [1] and Williams [2]. The present development further extends these earlier treatments to multiple temperatures and multicomponent thermal diffusion. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations which include the effects of ordinary, forced, pressure, and thermal diffusion. When thermal diffusion is neglected, these relations have the same form as the usual single-temperature ones, except that mole fractions are replaced by pressure fractions (i.e., ratios of partial pressures to total pressure). The binary and thermal diffusion coefficients are given in terms of collision integrals. Single-temperature systems and binary systems are treated as special cases of the general theory. A self-consistent effective binary diffusion approximation for multitemperature systems is presented.