The present invention discloses high selectivity copolyimide membranes for gas, vapor, and liquid separations. Gas permeation tests on these copolyimide membranes demonstrated that they not only showed high selectivity for CO2/CH4 separation, but also showed extremely high selectivities for H2/CH4 and He/CH4 separations. These copolyimide membranes can be used for a wide range of gas, vapor, and liquid separations such as separations of CO2/CH4, He/CH4, CO2/N2, olefin/paraffin separations (e.g. propylene/propane separation), H2/CH4, He/CH4, O2/N2, iso/normal paraffins, polar molecules such as H2O, H2S, and NH3 mixtures with CH4, N2, H2. The high selectivity copolyimide membranes have UV cross-linkable sulfonyl functional groups and can be used for the preparation of UV cross-linked high selectivity copolyimide membranes with enhanced selectivities. The invention also includes blend polymer membranes comprising the high selectivity copolyimide and polyethersulfone. The blend polymer membranes comprising the high selectivity copolyimide and polyethersulfone can be further UV cross-linked under UV radiation.