  We apply updated semi-analytic galaxy formation models simultaneously to the
stored halo/subhalo merger trees of the Millennium and Millennium-II
simulations. These differ by a factor of 125 in mass resolution, allowing
explicit testing of resolution effects on predicted galaxy properties. We have
revised the treatments of the transition between the rapid infall and cooling
flow regimes of gas accretion, of the sizes of bulges and of gaseous and
stellar disks, 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 5 orders of magnitude in stellar mass and 9
magnitudes in luminosity, but also to the observed abundance of Milky Way
satellites. This suggests that reionisation 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 ~30kpc
for massive galaxies. For M* < 6 x 10^10Msun, however, the model overpredicts
clustering at scales below 1 Mpc, suggesting that the sigma_8 adopted in the
simulations (0.9) is too high. Galaxy distributions within rich clusters agree
between the simulations and match those observed, but only if galaxies without
dark matter subhalos (so-called orphans) are included. Our model predicts a
larger passive fraction among low-mass galaxies than is observed, as well as an
overabundance of ~10^10Msun galaxies beyond z~0.6, reflecting deficiencies in
the way star-formation rates are modelled.
