Progress in membrane gas separation is impossible without the synthesis of new polymers with improved gas-transport and gas-separation characteristics. The most promising polymeric membrane materials with the advantageous combination of permeability and selectivity, which form the 2008 and 2015 Robeson "upper bounds," are polyheteroarylenes, among which are ladder polybenzodioxanes, polymers of intrinsic microporosity (PIM), polyimides, polyamides, and polyisathines. Their specific feature is the presence of moieties in the chemical structure that in any way contribute to the loosened packing of polymer chains and the increase in gas-permeability coefficients. Among such macromolecular design elements are groups with main chain kinks or bulky substituents increasing the rotation barriers and rigidity of macrochains. A high gas permeability of the polyheteroarylenes under consideration is commonly combined with an increased selectivity for many gas pairs (e.g., O-2/N-2, CO2/CH4) primarily associated with a high diffusion selectivity, which suggests their chain packing order and makes it possible to call them polymeric molecular sieves.