We show that nanometer-scale pores in single-layer freestanding graphene can effectively filter NaCl salt from water. Using classical molecular dynamics, we report the desalination performance of such membranes as a function of pore size, chemical functionalization, and applied pressure. Our results indicate that the membrane’s ability to prevent the salt passage depends critically on pore diameter with adequately sized pores allowing for water flow while blocking ions. Further, an investigation into the role of chemical functional groups bonded to the edges of graphene pores suggests that commonly occurring hydroxyl groups can roughly double the water flux thanks to their hydrophilic character. The increase in water flux comes at the expense of less consistent salt rejection performance, which we attribute to the ability of hydroxyl functional groups to substitute for water molecules in the hydration shell of the ions. Overall, our results indicate that the water permeability of this material is several orders of magnitude higher than conventional reverse osmosis membranes, and that nanoporous graphene may have a valuable role to play for water purification.