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The capture of CO2 is of high interest in our society representing an essential tool to mitigate man-made climate warming. Membrane technology applied for CO2 capture offers several advantages in terms of energy savings, simple operation, and easy scale-up. Glassy membranes are associated with low gas permeability that negatively affect on their industrial implementation. Oppositely, rubbery membranes offer high permeability, but their selectivity is low. Here we report rubbery organic frameworks (ROFs) combining the high permeability of soft matrices with the high sieving selectivity of molecular frameworks. The best performing membranes provide a CO2/N2 selectivity up to 104 with a CO2 permeability up to 1000 Barrer, representing relevant performances for industrial implementation. Water vapors have a positive effect on CO2 permeability, and the CO2/N2 selectivity is higher than in dry conditions, as most of CO2 gas emissions are present in fully humidified gas streams. The synergetic high permeability/selectivity performances are superior to that observed with current state-of-the-art polymeric membranes. [ABSTRACT FROM AUTHOR]