Biological membranes are exceptional materials that combine seemingly divergent properties. They are mechanically tough and difficult to rupture, yet they are highly fluid and readily change shape. They are permeable to certain molecules while impermeable to others. These unique properties make membranes an indispensable structural component of all living organisms and it has been proposed that life originated from simple protocell vesicles. These attributes also make membranes attractive from a materials perspective, leading to their use in diverse applications including drug delivery and biosensors. A materials scientist and a biological cell face similar challenges when using membranes to build materials or organelles. How can laterally heterogeneous compartmentalized membranes be designed? How can 3D membrane shape be dynamically manipulated? How can transport across membranes be regulated? We will elucidate design principles that govern these structures and processes, to enable engineering membrane-based materials and to illuminate how biological cells use membrane-based structures to achieve specific functions.