Gas separation by selective permeation through polymer membranes is one of the fastest growing branches of separation technology. Strong interest exists, therefore, in the synthesis of new polymers that exhibit both higher gas permeabilities and selectivities than presently available polymers. Such new polymers, in the form of asymmetric or "composite" membranes, are required in order to improve the economics of extant membrane processes for gas separations and to develop new processes. A considerable amount of information has been available for many years on the permeabilities and selectivities of a large variety of polymers to different gases. However, since most of these polymers were structurally unrelated, syntheses of new polymers for gas separations were based largely on trial and error and previous experience. It is only in recent years that the structure/permeability/selectivity relationships of polymers have become the object of systematic studies. The present review examines the progress made in the understanding of these relationships, with emphasis on selected rubbery and glassy polymers. Some of the most important theoretical models of gas transport in polymers are also reviewed. The potential usefulness of computer simulation techniques for predicting polymer structures that enhance penetrant gas mobility and selectivity is discussed. Finally, conjectures are offered as to possible advances in membrane separations of gaseous mixtures in the coming decade.