The emerging paradigm of 'abundant-data' computing requires real-time analytics on enormous quantities of data collected by a mushrooming network of sensors. Todays computing technology, however, cannot scale to satisfy such big data applications with the required throughput and energy efficiency. The next technology frontier will be monolithically integrated chips with three-dimensionally interleaved memory and logic for unprecedented data bandwidth with reduced energy consumption. In this work, we exploit the atomically thin nature of the graphene edge to assemble a resistive memory (∼ 3 Å thick) stacked in a vertical three-dimensional structure. We report some of the lowest power and energy consumption among the emerging non-volatile memories due to an extremely thin electrode with unique properties, low programming voltages, and low current. Circuit analysis of the three-dimensional architecture using experimentally measured device properties show higher storage potential for graphene devices compared that of metal based devices.