A large fraction of brown dwarfs and low-mass hydrogen-burning stars may form by gravitational fragmentation of protostellar discs. We explore the conditions for disc fragmentation and find that they are satisfied when a disc is large enough (≳100 au) so that its outer regions can cool efficiently, and it has enough mass to be gravitationally unstable, at such radii. We perform radiative hydrodynamic simulations and show that even a disc with mass 0.25 M⊙ and size 100 au fragments. The disc mass, radius and the ratio of disc to star mass (MD/M★≈ 0.36) are smaller than in previous studies. We find that fragmenting discs drastically decrease in mass and size within a few 104 yr of their formation, since a fraction of their mass, especially outside ∼100 au is consumed by the new stars and brown dwarfs that form. Fragmenting discs end up with masses ∼0.001–0.1 M⊙ and sizes ∼20–100 au. On the other hand, discs that are marginally stable evolve on a viscous time-scale, thus living longer (∼1–10 Myr). We produce simulated images of fragmenting discs and find that observing discs that are undergoing fragmentation is possible using current (e.g. IRAM Plateau de Bure interferometer) and future (e.g. ALMA) interferometers, but highly improbable due to the short duration of this process. Comparison with observations shows that many observed discs may be remnants of discs that have fragmented at an earlier stage. However, there are only a few candidates that are possibly massive and large enough to currently be gravitationally unstable. The rarity of massive (≳0.2 M⊙), extended (≳100 au) discs indicates that either such discs are highly transient (i.e. form, increase in mass becoming gravitationally unstable due to infall of material from the surrounding envelope and quickly fragment) or their formation is suppressed (e.g. by magnetic fields). We conclude that current observations of early-stage discs cannot exclude the mechanism of disc fragmentation.