This article covers the transport properties of gases, low activity vapors, and intermediate‐sized penetrants through rubbery, semicrystalline, and glassy polymers. The article emphasizes polymer transport properties under conditions of low to intermediate penetrant concentration where extraordinary differences can exist between the diffusivities of penetrants having relatively small differences in molecular size or shape. Nonideal transport effects, such as plasticization, clustering, and non‐Fickian transport, are covered. Particular attention is given to the nonequilibrium nature of glassy polymers and the resulting effects on penetrant transport. The recent development of positron annihilation lifetime spectroscopy (PALS) and molecular modeling of diffusion are described. Transport through porous polymers and reptation of macromolecules within a macromolecular matrix are not covered in this article.