Previous reports for pressure-driven transport through CNT membranes demonstrated CNT permeability values that were orders of magnitude larger than those predicted by Knudsen diffusion theory for gases (~102 fold enhancement) and by Hagen-Poiseuille equation for liquids (103-105 fold). While these results spurred great interest in CNTs for efficient membrane separations, it remains an open question if driving forces other than pressure could result in similar transport rate enhancements. A positive answer would greatly extend the promises and application space of CNTs as fluidic channels. In this work, we provide the first experimental evidence of enhanced gas transport in CNTs driven by a concentration rather than a pressure gradient. We fabricated cm2, free-standing, flexible, 1-5 nm SWNT/parylene membranes with well-aligned nanotubes as only transporting pores, and we measured the water vapor diffusion rate through the membrane when each surface is exposed to a different relative humidity. Our measurements demonstrate that these membranes exhibit rates of water vapor transport (~8000 gr/m2day) that surpass those of commercial breathable fabrics, even though the CNT pores are only a few nm wide and the overall porosity is less than 5.5%. Measured permeability of our CNT channels is 24 times larger than Knudsen diffusion prediction, and this flow enhancement is close to that measured for pressure-driven transport of nitrogen. Membranes made from 1-3 nm SWNT forests with higher number densities (> 1012/cm2) display even larger gas-transport enhancements. This ultrafast rate of water vapor transport in CNTs suggests that CNT membranes hold great potential for pervaporation, membrane distillation, and as building block of breathable and protective fabrics. For the last application, a membrane shall be able to block dangerous components while permitting perspiration. By demonstrating complete rejection of 3-nm charged dyes, 5-nm uncharged gold (Au) nanoparticles, and ~40-60-nm Dengue virus from aqueous solutions during filtration tests, we provide evidence that, in addition to outstanding breathability, our CNT membranes provide a high degree of protection from bio-threats by size exclusion.