Graphene is increasingly explored as a possible platform for developing novel separation technologies. This interest has arisen because it is a maximally thin membrane that, once perforated with atomic accuracy, may allow ultrafast and highly selective sieving of gases, liquids, dissolved ions and other species of interest. However, a perfect graphene monolayer is impermeable to all atoms and molecules under ambient conditions: even hydrogen, the smallest of atoms, is expected to take billions of years to penetrate graphene's dense electronic cloud. Only accelerated atoms possess the kinetic energy required to do this. The same behaviour might reasonably be expected in the case of other atomically thin crystals. Here we report transport and mass spectroscopy measurements which establish that monolayers of graphene and hexagonal boron nitride (hBN) are highly permeable to thermal protons under ambient conditions, whereas no proton transport is detected for thicker crystals such as monolayer molybdenum disulphide, bilayer graphene or multilayer hBN. Protons present an intermediate case between electrons (which can tunnel easily through atomically thin barriers) and atoms, yet our measured transport rates are unexpectedly high and raise fundamental questions about the details of the transport process. We see the highest room-temperature proton conductivity with monolayer hBN, for which we measure a resistivity to proton flow of about 10 Ω cm2 and a low activation energy of about 0.3 electronvolts. At higher temperatures, hBN is outperformed by graphene, the resistivity of which is estimated to fall below 10-3 Ω cm2 above 250 degrees Celsius. Proton transport can be further enhanced by decorating the graphene and hBN membranes with catalytic metal nanoparticles. The high, selective proton conductivity and stability make one-atom-thick crystals promising candidates for use in many hydrogen-based technologies.