Water transport through nanoporous materials is important in water treatment, desalination, and nanofiltration. Two-dimensional (2D) membranes such as porous graphene have been explored for high-permeance water transport. However, water transport through a new class of 2D membranes based on two-dimensional covalently linked fullerene monolayers has not been fully explored. Here we use classical molecular dynamics simulations to investigate both vapor and liquid water transport through a monolayer fullerene membrane. We find that a quasi-tetragonal phase fullerene membrane possesses the right pore size and geometry that allows fast water vapor transport (~ 50gm-2 day-1 Pa-1) and water liquid transport (~ 2.0gm-2 day-1 Pa-1). Furthermore, simulation of sea water transport through the fullerene membrane shows 100% salt rejection. The much faster vapor transport rate is attributed to the funnel-shaped pore and the optimal size that allows free rotation of water molecules permeating through, while the slower liquid transport is due to the need to desolvate a water molecule to break its hydrogen-bond network across the hydrophobic pore. This work shows the great potential of using monolayer fullerene membranes as 2D membranes for fast and selective water transport.