There is a critical need for new energy-efficient solutions to separate oil-water mixtures, especially those stabilized by surfactants. Traditional membrane-based separation technologies are energy-intensive and limited, either by fouling or by the inability of a single membrane to separate all types of oil-water mixtures. Here we report membranes with hygro-responsive surfaces, which are both superhydrophilic and superoleophobic, in air and under water. Our membranes can separate, for the first time, a range of different oil-water mixtures in a single-unit operation, with >99.9% separation efficiency, by using the difference in capillary forces acting on the two phases. Our separation methodology is solely gravity-driven and consequently is expected to be highly energy-efficient. We anticipate that our separation methodology will have numerous applications, including the clean-up of oil spills, wastewater treatment, fuel purification and the separation of commercially relevant emulsions.