We argue that brown dwarfs (BDs) and planemos form by the same mechanisms as low-mass hydrogen-burning stars, but that as one moves to lower and lower masses, an increasing fraction of these objects is formed by fragmentation of the outer parts (R ≳ 100 AU) of protostellar accretion discs around more massive primary protostars, which in turn formed in their own very-low-mass prestellar cores. Numerical simulations of disc fragmentation with realistic thermodynamics show that low-mass objects are readily formed by fragmentation of short-lived massive, extended protostellar accretion discs. Such objects tend subsequently to be liberated into the field at low speed, due to mutual interactions with the primary protostar. Many (~20%) are in low-mass (M1 + M2 < 0.2M⊙) binary systems with semi-major axes a ~ 1 to 2 AU or ~200 AU and mass ratios q ≡ M2/M1 ≳ 0.7. Most of the brown dwarfs have sufficiently large attendant discs to sustain accretion and outflows. Most of the BDs that remain bound to the primary protostar have wide orbits (i.e., there is a BD desert), and these BDs also have a significantly higher probability of being in a BD/BD binary system than do the brown dwarfs that are liberated into the field (just as observed). In this picture, the multiplicity statistics and velocity dispersion of brown dwarfs are largely determined by the eigen evolution of a small-N system, born from a single prestellar core, rather than the larger-scale dynamics of the parent cluster. Consequently, many of the statistical properties of brown dwarfs should not differ very much from one star-formation region to another.