Wide attention has been paid in recent years to diffusion through polymeric materials for its interesting applications in various technological fields [91Vie]. Examples are permselective membranes for separation of gases, vapors and liquids, food packing, protective coatings and controlled drug delivery systems. Moreover, diffusion studies have provided valuable information on the structure of polymers and timedependent changes in the glassy state. Particularly gas molecules were often used as structure-sensitive probes of variable size.Since this volume is concerned with diffusion in solids, the present chapter covers diffusion in glassy polymers and semicrystalline polymers with a glassy amorphous phase. The diffusion behavior of semicrystalline polymers is essentially governed by the amorphous phase. For crystalline polymers the reader is referred to Chap. 8 on molecular solids.Diffusion characteristics of polymers can be markedly affected by the presence of additives, such as plasticizers and stabilizers, and by residual solvent. In glassy polymers the polymer molecular weight generally has no influence unless it is very low [82Toi2]. The non-equilibrium nature of the glassy state gives rise to a history dependence of transport properties and the appearance of local inhomogenities [91Vie, 94Pet, 89Kor]. The latter are produced by the residual mobility of different frozen chains, which are unable to relax to thermal equilibrium because of inhibited segmental motions. The influence of history on current and future performance of glassy polymers is well documented for thermal and mechanical treatment as well as for processing conditions. Furthermore, many studies have reported penetrant-induced history-dependent effects, often termed conditioning in connection with transport in glassy polymers. History also affects the orientation of molecules and the crystallinity of semicrystalline polymers. History effects manifest themselfs in time-dependent and hysteretic behavior. For instance, structural relaxation of glassy polymers, which is accompanied by annealing of excess volume, may lead to a marked decrease of penetrant diffusivities [90Ehl, 94Zha]. On experimental time scales diffusivities finally approach a time-independent value being characteristic of the well relaxed glassy state. An example of penetrant-induced conditioning and hysteresis is the exposure of glassy polymers to relatively high CO 2 pressures, which induces a volumetric expansion effect. A substantial increase in excess volume persists long after the penetrant has been removed. No hysteretic effects are usually observed, even at high pressure in the MPa range, for gases such as He, N 2 and even CH 4 [89Kor]. The lower condensability, and hence lower solubility of these supercritical gases, is presumable responsible for their lack of conditioning capability.In view of the abovementioned complications it is obvious that diffusion data in glassy polymers are often not very meaningful without detailed information on sample preparation and history. ...