Dialysis is a ubiquitous separation process in biochemical processing and biological research. State-of-the-art dialysis membranes comprise a relatively thick polymer layer with tortuous pores, and suffer from low rates of diffusion leading to extremely long process times (often several days) and poor selectivity, especially in the 0-1000 Da molecular weight cut-off range. Here, the fabrication of large-area (cm ) nanoporous atomically thin membranes (NATMs) is reported, by transferring graphene synthesized using scalable chemical vapor deposition (CVD) to polycarbonate track-etched supports. After sealing defects introduced during transfer/handling by interfacial polymerization, a facile oxygen-plasma etch is used to create size-selective pores (≤1 nm) in the CVD graphene. Size-selective separation and desalting of small model molecules (≈200-1355 Da) and proteins (≈14 000 Da) are demonstrated, with ≈1-2 orders of magnitude increase in permeance compared to state-of-the-art commercial membranes. Rapid diffusion and size-selectivity in NATMs offers transformative opportunities in purification of drugs, removal of residual reactants, biochemical analytics, medical diagnostics, therapeutics, and nano-bio separations.