The present invention discloses high performance polybenzoxazole membranes prepared from aromatic poly(o-hydroxy amide) membranes by thermal cyclization and a method for using these membranes. The polybenzoxazole membranes were prepared by thermal treating aromatic poly(o-hydroxy amide) membranes in a temperature range of 200° to 550° C. under inert atmosphere. The aromatic poly(o-hydroxy amide) membranes used for making the polybenzoxazole membranes were prepared from aromatic poly(o-hydroxy amide) polymers comprising pendent phenolic hydroxyl groups ortho to the amide nitrogen in the polymer backbone. In some embodiments of the invention, the polybenzoxazole membranes may be subjected to an additional crosslinking step to increase the selectivity of the membranes. These polybenzoxazole membranes showed significantly improved permeability for gas separations compared to the precursor aromatic poly(o-hydroxy amide) membranes and are not only suitable for a variety of liquid, gas, and vapor separations, but also can be used in catalysis and fuel cells.