We use the combination of the very large Millennium and Millennium-II simulations to follow galaxy formation over a stellar mass range of more than five orders of magnitude throughout representative cosmological volumes. These are the first simulations to address the abundance and spatial distribution of galaxies ranging from cD’s down to dSph’s. We demonstrate that stronger feedback is needed than in previous models, epically at low mass, if the SDSS stellar mass function is to be matched in a LCDM universe. Models which satisfy this constraint also reproduce the observed abundance of faint dwarfs around the Milky Way and M 31. Only the very faintest objects are significantly affected by the treatment of reionisation. Our model can reproduce most galaxy properties in the local Universe. Problems remain with respect to the stellar age distributions and kinematics of dwarfs.