Recent theoretical and experimental studies reported ultra-high water permeability and salt rejection in nanoporous single-layer graphene. However, creating and controlling the size and distribution of nanometer-scale pores pose significant challenges to application of these membranes for water desalination. Graphyne and hydrogenated graphyne have tremendous potential as ultra-permeable membranes for desalination and wastewater reclamation due to their uniform pore-distribution, atomic thickness and mechano-chemical stability. Using molecular dynamics (MD) simulations and upscale continuum analysis, the desalination performance of bare and hydrogenated α-graphyne and γ-{2,3,4}-graphyne membranes is evaluated as a function of pore size, pore geometry, chemical functionalization and applied pressure. MD simulations show that pores ranging from 20 to 50 Ų reject in excess of 90% of the ions for pressures up to 1 GPa. Water permeability is found to range up to 85 L cm⁻² day⁻¹ MPa⁻¹, which is up to three orders of magnitude larger than commercial seawater reverse osmosis (RO) membranes and up to ten times that of nanoporous graphene. Pore chemistry, functionalization and geometry are shown to play a critical role in modulating the water flux, and these observations are explained by water velocity, density, and energy barriers in the pores. The atomistic scale investigations are complemented by upscale continuum analysis to examine the performance of these membranes in application to cross-flow RO systems. This upscale analysis, however, shows that the significant increase in permeability, observed from MD simulations, does not fully translate to current RO systems due to transport limitations. Nevertheless, upscale calculations predict that the higher permeability of graphyne membranes would allow up to six times higher permeate recovery or up to 6% less energy consumption as compared to thin-film composite membranes at currently accessible operating conditions. Significantly higher energy savings and permeate recovery can be achieved if higher feed-flow rates can be realized.