Molecular transport and quantum tunneling of H2 and H2O molecules through nanoporous gra-phene is studied using computational modeling and first principles density functional theory. It is demonstrated that molecules with sufficiently high kinetic energies can tunnel through the na-nopores. It is also demonstrated that molecules can be trapped in front of the nanopore or behind it. These investigations help us learn the behavior of molecules in and around nanopores of gra-phene. They also help us learn fundamentals of molecular tunneling. We believe, nanoporous gra-phene can play important roles for gas separation and nanofiltration.