The memristor(1,2) is a promising building block for next-generation non-volatile memory(3), artificial neural networks(4-7) and bio-inspired computing systems(8,9). Organizing small memristors into high-density crossbar arrays is critical to meet the ever-growing demands in high-capacity and low-energy consumption, but this is challenging because of difficulties in making highly ordered conductive nanoelectrodes. Carbon nanotubes, graphene nanoribbons and dopant nanowires have potential as electrodes for discrete nanodevices(10-14), but unfortunately these are difficult to pack into ordered arrays. Transfer printing, on the other hand, is effective in generating dense electrode arrays(15) but has yet to prove suitable for making fully random accessible crossbars. All the aforementioned electrodes have dramatically increased resistance at the nanoscale(16-18), imposing a significant barrier to their adoption in operational circuits. Here we demonstrate memristor crossbar arrays with a 2-nm feature size and a single-layer density up to 4.5 terabits per square inch, comparable to the information density achieved using three-dimensional stacking in state-of-the-art 64-layer and multilevel 3D-NAND flash memory(19). Memristors in the arrays switch with tens of nanoamperes electric current with nonlinear behaviour. The densely packed crossbar arrays of individually accessible, extremely small functional memristors provide a power-efficient solution for information storage and processing.