Combining the electronic properties of graphene and molybdenum disulphide (MoS2) in hybrid heterostructures offers the possibility to create devices with various functionalities. Electronic logic and memory devices have already been constructed from graphene-MoS2 hybrids, but they do not make use of the photosensitivity of MoS2, which arises from its optical-range bandgap. Here, we demonstrate that graphene-on-MoS2 binary heterostructures display remarkable dual optoelectronic functionality, including highly sensitive photodetection and gate-tunable persistent photoconductivity. The responsivity of the hybrids was found to be nearly 1 × 10(10) A W(-1) at 130 K and 5 × 10(8) A W(-1) at room temperature, making them the most sensitive graphene-based photodetectors. When subjected to time-dependent photoillumination, the hybrids could also function as a rewritable optoelectronic switch or memory, where the persistent state shows almost no relaxation or decay within experimental timescales, indicating near-perfect charge retention. These effects can be quantitatively explained by gate-tunable charge exchange between the graphene and MoS2 layers, and may lead to new graphene-based optoelectronic devices that are naturally scalable for large-area applications at room temperature.