We suggest that a high proportion of brown dwarfs are formed by gravitational fragmentation of massive, extended discs around Sun-like stars. We argue that such discs should arise frequently, but should be observed infrequently, precisely because they fragment rapidly. By performing an ensemble of radiation-hydrodynamic simulations, we show that such discs typically fragment within a few thousand years to produce mainly brown dwarfs (including planetary-mass brown dwarfs) and low-mass hydrogen-burning stars. Subsequently most of the brown dwarfs are ejected by mutual interactions. We analyse the properties of these objects that form by disc fragmentation, and compare them with observations.