Polymersomes are self-assembled polymer shells composed of block copolymer amphiphiles. These synthetic amphiphiles have amphiphilicity similar to lipids, but they have much larger molecular weights, so for this reason--along with others reviewed here--comparisons of polymersomes with viral capsids composed of large polypeptide chains are highly appropriate. We summarize the wide range of polymers used to make polymersomes along with descriptions of physical properties such as stability and permeability. We also elaborate on emerging studies of in vivo stealthiness, programmed disassembly for controlled release, targeting in vitro, and tumor-shrinkage in vivo. Comparisons of polymersomes with viral capsids are shown to encompass and inspire many aspects of current designs.