As a result of quantum-confinement effects, the emission colour of semiconductor nanocrystals can be modified dramatically by simply changing their size. Such spectral tunability, together with large photoluminescence quantum yields and high photostability, make nanocrystals attractive for use in a variety of light-emitting technologies--for example, displays, fluorescence tagging, solid-state lighting and lasers. An important limitation for such applications, however, is the difficulty of achieving electrical pumping, largely due to the presence of an insulating organic capping layer on the nanocrystals. Here, we describe an approach for indirect injection of electron-hole pairs (the electron-hole radiative recombination gives rise to light emission) into nanocrystals by non-contact, non-radiative energy transfer from a proximal quantum well that can in principle be pumped either electrically or optically. Our theoretical and experimental results indicate that this transfer is fast enough to compete with electron-hole recombination in the quantum well, and results in greater than 50 per cent energy-transfer efficiencies in the tested structures. Furthermore, the measured energy-transfer rates are sufficiently large to provide pumping in the stimulated emission regime, indicating the feasibility of nanocrystal-based optical amplifiers and lasers based on this approach.