Graphene is an attractive material for optoelectronics¹ and photodetection applications^2,3,4,5,6 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 ∼10⁻² A W⁻¹. Here, we demonstrate a gain of ∼10⁸ electrons per photon and a responsivity of ∼10⁷ A W⁻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 7 × 10¹³ Jones, benefits from gate-tunable sensitivity and speed, spectral selectivity from the short-wavelength infrared to the visible, and compatibility with current circuit technologies.