This invention relates to self-cross-linkable and self-cross-linked aromatic polyimide polymers, their membranes and methods for making and using these polymers and membranes. The self-cross-linkable aromatic polyimide polymer described in the present invention comprises both hydroxyl functional groups and carboxylic acid functional groups. The self-cross-linked aromatic polyimide was formed via heating the self-cross-linkable aromatic polyimide polymer at ≦300° C. The self-cross-linked aromatic polyimide membranes exhibit high selectivity in separation of mixtures of gases and liquids.