We have updated and extended our semi-analytic galaxy formation modelling capabilities and applied them simultaneously to the stored halo/subhalo merger trees of the Millennium and Millennium-II Simulations (MS and MS-II, respectively). These differ by a factor of 125 in mass resolution, allowing explicit testing of resolution effects on predicted galaxy properties. We have revised the treatment of the transition between the rapid infall and cooling flow regimes of gas accretion, of the sizes of bulges, and of gaseous and stellar discs, of supernova feedback, of the transition between central and satellite status as galaxies fall into larger systems, and of gas and star stripping, once they become satellites. Plausible values of efficiency and scaling parameters yield an excellent fit not only to the observed abundance of low-redshift galaxies over five orders of magnitude in stellar mass and 9 mag in luminosity, but also to the observed abundance of Milky Way satellites. This suggests that reionization effects may not be needed to solve the ‘missing-satellite’ problem, except, perhaps, for the faintest objects. The same model matches the observed large-scale clustering of galaxies as a function of stellar mass and colour. The fit remains excellent down to ∼30 kpc for massive galaxies. For M* < 6 × 1010 M⊙, however, the model overpredicts clustering at scales below ∼1 Mpc, suggesting that the assumed fluctuation amplitude, σ8= 0.9, is too high. The observed difference in clustering between active and passive galaxies is matched quite well for all masses. Galaxy distributions within rich clusters agree between the simulations and match those observed, but only if galaxies without dark matter subhaloes (so-called orphans) are included. Even at MS-II resolution, schemes which assign galaxies only to resolved dark matter subhaloes cannot match observed clusters. Our model predicts a larger passive fraction among low-mass galaxies than is observed, as well as an overabundance of ∼1010 M⊙ galaxies beyond z∼ 0.6. (The abundance of ∼1011 M⊙ galaxies is matched out to z∼ 3.) These discrepancies appear to reflect deficiencies in the way star formation rates are modelled.