Graphene and other two-dimensional materials offer a new class of ultrathin membranes that can have atomically defined nanopores with diameters approaching those of hydrated ions(1-7). These nanopores have the smallest possible pore volumes of any ion channel, which, due to ionic dehydration(8) and electrokinetic effects9, places them in a novel transport regime and allows membranes to be created that combine selective ionic transport(10) with ultimate permeance(11-13) and could lead to separations(14,15) and sensing(16) applications. However, experimental characterization and understanding of sub-continuum ionic transport in nanopores below 2 nm is limited(17,18). Here we show that isolated sub-2 nm pores in graphene exhibit, in contrast to larger pores, diverse transport behaviours consistent with ion transport over a free-energy barrier arising from ion dehydration and electrostatic interactions. Current-voltage measurements reveal that the conductance of graphene nanopores spans three orders of magnitude(8) and that they display distinct linear, voltage-activated or rectified current-voltage characteristics and different cation-selectivity profiles. In rare cases, rapid, voltage-dependent stochastic switching is observed, consistent with the presence of a dissociable group in the pore vicinity(19). A modified Nernst-Planck model incorporating ion hydration and electrostatic effects quantitatively matches the observed behaviours.