We study the rate at which gas accretes on to galaxies and haloes and investigate whether the accreted gas was shocked to high temperatures before reaching a galaxy. For this purpose, we use a suite of large cosmological, hydrodynamical simulations from the OverWhelmingly Large Simulations project, which uses a modified version of the smoothed particle hydrodynamics code GADGET-3. We improve on previous work by considering a wider range of halo masses and redshifts, by distinguishing between accretion on to haloes and accretion on to galaxies, by including important feedback processes and by comparing simulations with different physics. Gas accretion is mostly smooth, with mergers only becoming important for groups and clusters. The specific rate of the gas accretion on to haloes is, like that for dark matter, only weakly dependent on the halo mass. For halo masses Mhalo≫ 1011 M⊙, it is relatively insensitive to feedback processes. In contrast, accretion rates on to galaxies are determined by radiative cooling and by outflows driven by supernovae and active galactic nuclei. Galactic winds increase the halo mass at which the central galaxies grow the fastest by about two orders of magnitude to Mhalo˜ 1012 M⊙. Gas accretion is bimodal, with maximum past temperatures either of the order of the virial temperature or ≲105 K. The fraction of the gas accreted on to haloes in the hot mode is insensitive to feedback and metal-line cooling. It increases with decreasing redshift, but is mostly determined by the halo mass, increasing gradually from less than 10 per cent for ˜1011 M⊙ to greater than 90 per cent at ˜1013 M⊙. In contrast, for accretion on to galaxies, the cold mode is always significant and the relative contributions of the two accretion modes are more sensitive to feedback and metal-line cooling. On average, the majority of stars present in any mass halo at any redshift were formed from the gas accreted in the cold mode, although the hot mode contributes typically over 10 per cent for Mhalo≳ 1011 M⊙. Thus, while gas accretion on to haloes can be robustly predicted, the rate of accretion on to galaxies is sensitive to uncertain feedback processes. Nevertheless, it is clear that galaxies, but not necessarily their gaseous haloes, are predominantly fed by the gas that did not experience an accretion shock when it entered the host halo.