Vesicles are microscopic sacs that enclose a volume with a molecularly thin membrane. The membranes are generally self-directed assemblies of amphiphilic molecules with a dual hydrophilic-hydrophobic character. Biological amphiphiles form vesicles central to cell function and are principally lipids of molecular weight less than 1 kilodalton. Block copolymers that mimic lipid amphiphilicity can also self-assemble into vesicles in dilute solution, but polymer molecular weights can be orders of magnitude greater than those of lipids. Structural features of vesicles, as well as properties including stability, fluidity, and intermembrane dynamics, are greatly influenced by characteristics of the polymers. Future applications of polymer vesicles will rely on exploiting unique property-performance relations, but results to date already underscore the fact that biologically derived vesicles are but a small subset of what is physically and chemically possible.