Membranes with ~1 nanometer size-based selectivity have numerous applications in the biochemical and pharmaceutical industries, as well as wastewater treatment processes. However, most commercial membranes show poor selectivity in this size range due to wide pore size distributions, and charged surfaces that complicate the separation mechanisms involved. There is a need for new membrane materials that enable these separations while exhibiting high flux and fouling resistance, and are readily fabricated into large-area membranes. Zwitterionic groups strongly resist biomacromolecular fouling due to their high degree of hydration, which makes them promising materials for membrane applications. Zwitterions are also documented to self-assemble into channel-type clusters 0.6-2 nm in size. We introduce a new class of membranes whose selective layers are made of zwitterionic amphiphilic random copolymers. These membranes derive their permeability, selectivity and fouling resistance from this self-assembled nanostructure. We synthesized random copolymers of hydrophobic monomers (acrylonitrile, methyl methacrylate, trifluoro ethyl methacrylate) and zwitterionic monomers (sulfobetaine methacrylate, sulfobetaine vinylpyridines, and phosphorylcholine) by free radical polymerization. These copolymers self-assemble into bicontinuous networks of ~1 nm nanochannels, driven by the strong dipoles of the zwitterionic groups, and documented by transmission electron microscopy (TEM) imaging. Membranes prepared by coating these polymers on porous supports exhibit high flux, size-based selectivity with a ~1 nm size cut-off, and excellent fouling resistance to common foulants. We have systematically investigated how copolymer composition (monomer structure, ratio) affects nanostructure (self-assembled domain size) and the membrane performance. We expect these membranes to be promising candidates for textile wastewater treatment, pharmaceutical purification and bioseparation applications.