NOVELTY - Polybenzoxazole membrane (I) prepared from an aromatic poly(o-hydroxy amide) polymer, is claimed.    USE - (I) is useful for separating at least one gas or liquid from a mixture of gases or liquids, where the separation is desalination of water by reverse osmosis, non-aqueous liquid separation, ethanol/water separations, pervaporation dehydration of aqueous/organic mixtures, carbon dioxide (CO2)/methane (CH4), CO2/nitrogen (N2), hydrogen/CH4, oxygen/N2, hydrogen sulfide/CH4, olefin/paraffin and iso/normal paraffins separations (all claimed). (I) is useful for catalysis and fuel cells applications. (I) is useful in gas/vapor separation processes in chemical, petrochemical, pharmaceutical and allied industries for removing organic vapors from gas streams, e.g.: in off-gas treatment for recovery of volatile organic compounds; in hydrocarbon vapor separation from hydrogen in oil and gas refineries; for hydrocarbon dew pointing of natural gas (i.e. to decrease the hydrocarbon dew point to below the lowest possible export pipeline temperature so that liquid hydrocarbons do not separate in the pipeline); for control of methane number in fuel gas for gas engines and gas turbines; and for gasoline recovery. (I) is useful in: the separation of liquid mixtures by pervaporation, such as in the removal of organic compounds (e.g. alcohols phenols, chlorinated hydrocarbons, pyridines and ketones) from water such as aqueous effluents or process fluids; and the separation of mixtures of organic compounds e.g. ethyl acetate-ethanol, diethyl ether-ethanol, benzene-ethanol, chloroform-ethanol, butanol-butylacetate and ethyl acetate-ethanol-acetic acid. (I) is useful: for separation of organic molecules from water (e.g. ethanol and/or phenol from water) and removal of metal and other organic compounds from water; for the separation of gas mixtures including carbon dioxide removal from natural gas; to concentrate olefins in a paraffin/olefin stream for olefin cracking applications e.g. for propylene/propane separation to increase the concentration of the effluent in a catalytic dehydrogenation reaction for the production of propylene from propane and isobutylene from isobutane; and as the separator in chemical reactors to enhance the yield of equilibrium-limited reactions by selective removal of a specific substance.    ADVANTAGE - (I) exhibits improved permeability for gas separations compared to the precursor aromatic poly(o-hydroxy amide) membranes. (I) has ease of processability, high mechanical stability, high selectivity, high permeation rate or flux, high thermal stability, high chemical stability, and stable flux and sustained selectivity over time by resistance to solvent swelling, plasticization and hydrocarbon contaminants.    DETAILED DESCRIPTION - INDEPENDENT CLAIMS are included for:    (1) preparing (I) comprising subjecting an aromatic poly(o-hydroxy amide) membrane prepared from an aromatic poly(o-hydroxyamide) polymer to thermal cyclization at 200-550 degrees C under an inert atmosphere; and    (2) separating at least one gas or liquid from a mixture of gases or liquids comprising providing (I) prepared from aromatic poly(o-hydroxy amide) membrane that is permeable to one gas or liquid; contacting the mixture of gases or liquids on one side of (I) to cause one gas or liquid to permeate (I); and removing from the opposite side of the membrane a permeate gas or liquid composition that is a portion of the one gas or liquid which permeated the membrane.