Pervaporation membranes including poly(acrylamide)-grafted alginate membranes, which may be optimized for the separation of alcohols from water, low viscosity sodium alginate membranes containing PEG and PVA, which may be optimized for the separation of organic acids from water, and copolymeric PAN-grafted PVA membranes, which may be optimized for the separation of DMF from water, and methods of making such membranes. Use of such membranes in pervaporation and pervaporation devices containing such membranes. Use of such membranes alone or in combination with ion-exchange membranes for recovery of organic compounds or for water purification applications such as production of potable water or industrial waste treatment. The membranes of the present invention may be used to remove trace amounts of water from organic compounds.