This abstract discusses membranes needed to separate fluids for the production of oxygen-enriched air, nitrogen-enriched-air, for the separation of carbon dioxide from hydrocarbons, and the separation of hydrogen from various petrochemical and oil refining streams. Membranes are needed that provide a resistance to interaction with process components or contaminants, provide the mechanical strength required to withstand high membrane differential pressures and high process temperatures, and exhibit sufficient maximum strain such that membranes are not brittle and can easily be formed into desirable membrane forms. Membranes of polyimide polymers, particularly polyimide polymers sold under the trade name P-84, are annealed in a controlled annealing step to improve the mechanical properties of the polymers used to make separation membranes. The resulting annealed polyimide polymer is used to produce various forms of high strength, chemically resistant membranes, including hollow-fiber membranes that are suitable for high pressure, high temperature applications.