We use galaxy and dark halo data from the public database for the Millennium Simulation to study the growth of galaxies in the De Lucia et al. model for galaxy formation. Previous work has shown this model to reproduce many aspects of the systematic properties and the clustering of real galaxies, both in the nearby universe and at high redshift. It assumes the stellar masses of galaxies to increase through three processes, major mergers, the accretion of smaller satellite systems and star formation. We show the relative importance of these three modes to be a strong function of stellar mass and redshift. Galaxy growth through major mergers depends strongly on stellar mass, but only weakly on redshift. Except for massive systems, minor mergers contribute more to galaxy growth than major mergers at all redshifts and stellar masses. For galaxies significantly less massive than the Milky Way, star formation dominates the growth at all epochs. For galaxies significantly more massive than the Milky Way, growth through mergers is the dominant process at all epochs. At a stellar mass of 6 × 1010 M⊙, about that of the Milk Way, star formation dominates at z > 1 and mergers at later times. At every stellar mass, the growth rates through star formation increase rapidly with increasing redshift. Specific star formation rates are the decreasing function of stellar mass not only at z= 0 but also at all higher redshifts. For comparison, we carry out a similar analysis of the growth of dark matter haloes. In contrast to the galaxies, growth rates depend strongly on redshift, but only weakly on mass. They agree qualitatively with analytic predictions for halo growth.