Continued progress in high-speed computing depends on breakthroughs in both materials synthesis and device architectures(1-4). The performance of logic and memory can be enhanced significantly by introducing a memristor(5,6), a two-terminal device with internal resistance that depends on the history of the external bias voltage(5-7). State-of-the-art memristors, based on metal-insulator-metal (MIM) structures with insulating oxides, such as TiO2, are limited by a lack of control over the filament formation and external control of the switching voltage(3,4,6,8,9). Here, we report a class of memristors based on grain boundaries (GBs) in single-layer MoS2 devices(10-12). Specifically, the resistance of GBs emerging from contacts can be easily and repeatedly modulated, with switching ratios up to similar to 10(3) and a dynamic negative differential resistance (NDR). Furthermore, the atomically thin nature of MoS2 enables tuning of the set voltage by a third gate terminal in a field-effect geometry, which provides new functionality that is not observed in other known memristive devices.