We report results from radiation hydrodynamical simulations of the collapse of molecular cloud cores to form protostars. The calculations follow the formation and evolution of the first hydrostatic core/disc, the collapse to form a stellar core, and effect of stellar core formation on the surrounding disc and envelope. Past barotropic calculations have shown that rapidly rotating first cores evolve into ‘pre-stellar discs’ with radii up to ∼100 au that may last thousands of years before a stellar core forms. We investigate how the inclusion of a realistic equation of state and radiative transfer alters this behaviour, finding that the qualitative behaviour is similar, but that the pre-stellar discs may last 1.5–3 times longer in the more realistic calculations. The masses, radii and lifetimes of the discs increase for initial molecular cloud cores with faster rotation rates. In the most extreme case we model, a pre-stellar disc with a mass of 0.22 M⊙ and a radius of ≈100 au can form in a 1-M⊙ cloud and last several thousand years before a stellar core is formed. Such large, massive objects may be imaged using the Atacama Large Millimeter/Submillimeter Array. Fragmentation of these massive discs may also provide an effective route to binary and multiple star formation, before radiative feedback from accretion on to the stellar core can inhibit fragmentation. Once collapse to form a stellar core occurs within the pre-stellar disc, the radiation hydrodynamical simulations produce qualitatively different behaviour from the barotropic calculations due to the accretion energy released. This drives a shockwave through the circumstellar disc and launches a bipolar outflow even in the absence of magnetic fields.