Graphene is an attractive material for optoelectronics(1) and photodetection applications(2-6) because it offers a broad spectral bandwidth and fast response times. However, weak light absorption and the absence of a gain mechanism that can generate multiple charge carriers from one incident photon have limited the responsivity of graphene-based photodetectors to similar to 10(-2) AW(-1). Here, we demonstrate a gain of similar to 10(8) electrons per photon and a responsivity of similar to 10(7) AW(-1) in a hybrid photodetector that consists of monolayer or bilayer graphene covered with a thin film of colloidal quantum dots. Strong and tunable light absorption in the quantum-dot layer creates electric charges that are transferred to the graphene, where they recirculate many times due to the high charge mobility of graphene and long trapped-charge lifetimes in the quantum-dot layer. The device, with a specific detectivity of 7x10(13) Jones, benefits from gate-tunable sensitivity and speed, spectral selectivity from the short-wavelength infrared to the visible, and compatibility with current circuit technologies.