We use time-dependent, one-dimensional disc models to investigate the evolution of protostellar discs that form through the collapse of molecular cloud cores and in which the primary transport mechanism is self-gravity. We assume that these discs settle into a state of thermal equilibrium with Q = 2 and that the strength of the angular momentum transport is set by the cooling rate of the disc. The results suggest that these discs will attain a quasi-steady state that persists for a number of free-fall times and in which most of the mass within 100au is located inside 10-20au. This pile-up of mass in the inner disc could result in temperatures that are high enough for the growth of magnetohydrodynamic turbulence which could rapidly drain the inner disc and lead to FU Orionis-like outbursts. In all our simulations, the inner regions of the discs (r < 40au) were stable against fragmentation,while fragmentation was possible in the outer regions (r > 40au) of discs that formed from cores that had enough initial angular momentum to deposit sufficient mass in these outer regions. The large amount of mass in these outer regions, however, suggests that fragmentation will lead to the formation of sub-stellar and stellar mass companions, rather than planetary mass objects. Although mass accretion rates were largely consistent with observations, the large disc masses suggest that an additional transport mechanism (such as magnetorotational instability occurring in the upper layers of the disc) must operate in order to drain the remaining disc material within observed disc lifetimes.