The present invention provides high permeance copolyimide membranes and methods for making and using these membranes for gas separations such as for hydrogen purification and for acid gas removal from natural gas. The random copolyimide polymers used to make the copolyimide membrane may be UV crosslinked to improve selectivity in separating mixtures of gases or in purifying liquids. The membranes may be fabricated into any known membrane configuration such as a flat sheet or a hollow fiber.