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Increasing demands for energy-efficient separations in applications ranging from water purification to petroleum refining, chemicals production, and carbon capture have stimulated a vigorous search for novel, high-performance separation membranes. Synthetic membranes suffer a ubiquitous, pernicious trade-off: highly permeable membranes lack selectivity and vice versa. However, materials with both high permeability and high selectivity are beginning to emerge. For example, design features from biological membranes have been applied to break the permeability-selectivity trade-off. We review the basis for the permeability-selectivity trade-off, state-of-the-art approaches to membrane materials design to overcome the trade-off, and factors other than permeability and selectivity that govern membrane performance and, in turn, influence membrane design.