BACKGROUND Synthetic membranes are used for desalination, dialysis, sterile filtration, food processing, dehydration of air and other industrial, medical, and environmental applications due to low energy requirements, compact design, and mechanical simplicity. New applications are emerging from the water-energy nexus, shale gas extraction, and environmental needs such as carbon capture. All membranes exhibit a trade-off between permeability—i.e., how fast molecules pass through a membrane material—and selectivity—i.e., to what extent the desired molecules are separated from the rest. However, biological membranes such as aquaporins and ion channels are both highly permeable and highly selective. Separation based on size difference is common, but there are other ways to either block one component or enhance transport of another through a membrane. Based on increasing molecular understanding of both biological and synthetic membranes, key design criteria for new membranes have emerged: (i) properly sized free-volume elements (or pores), (ii) narrow free-volume element (or pore size) distribution, (iii) a thin active layer, and (iv) highly tuned interactions between permeants of interest and the membrane. Here, we discuss the permeability/selectivity trade-off, highlight similarities and differences between synthetic and biological membranes, describe challenges for existing membranes, and identify fruitful areas of future research. ADVANCES Many organic, inorganic, and hybrid materials have emerged as potential membranes. In addition to polymers, used for most membranes today, materials such as carbon molecular sieves, ceramics, zeolites, various nanomaterials (e.g., graphene, graphene oxide, and metal organic frameworks), and their mixtures with polymers have been explored. Simultaneously, global challenges such as climate change and rapid population growth stimulate the search for efficient water purification and energy-generation technologies, many of which are membrane-based. Additional driving forces include wastewater reuse from shale gas extraction and improvement of chemical and petrochemical separation processes by increasing the use of light hydrocarbons for chemicals manufacturing. OUTLOOK Opportunities for advancing membranes include (i) more mechanically, chemically, and thermally robust materials; (ii) judiciously higher permeability and selectivity for applications where such improvements matter; and (iii) more emphasis on fundamental structure/property/processing relations. There is a pressing need for membranes with improved selectivity, rather than membranes with improved permeability, especially for water purification. Modeling at all length scales is needed to develop a coherent molecular understanding of membrane properties, provide insight for future materials design, and clarify the fundamental basis for trade-off behavior. Basic molecular-level understanding of thermodynamic and diffusion properties of water and ions in charged membranes for desalination and energy applications such as fuel cells is largely incomplete. Fundamental understanding of membrane structure optimization to control transport of minor species (e.g., trace-organic contaminants in desalination membranes, neutral compounds in charged membranes, and heavy hydrocarbons in membranes for natural gas separation) is needed. Laboratory evaluation of membranes is often conducted with highly idealized mixtures, so separation performance in real applications with complex mixtures is poorly understood. Lack of systematic understanding of methodologies to scale promising membranes from the few square centimeters needed for laboratory studies to the thousands of square meters needed for large applications stymies membrane deployment. Nevertheless, opportunities for membranes in both existing and emerging applications, together with an expanding set of membrane materials, hold great promise for membranes to effectively address separations needs.