We use cosmological hydrodynamical simulations to investigate how the inclusion of physical processes relevant to galaxy formation (star formation, metal-line cooling, stellar winds, supernovae and feedback from active galactic nuclei, AGN) change the properties of haloes, over four orders of magnitude in mass. We find that gas expulsion and the associated dark matter (DM) expansion induced by supernova-driven winds are important for haloes with masses M200 ≲ 1013 M⊙, lowering their masses by up to 20 per cent relative to a DM-only model. AGN feedback, which is required to prevent overcooling, has a significant impact on halo masses all the way up to cluster scales (M200 ˜ 1015 M⊙). Baryon physics changes the total mass profiles of haloes out to several times the virial radius, a modification that cannot be captured by a change in the halo concentration. The decrease in the total halo mass causes a decrease in the halo mass function of about 20 per cent. This effect can have important consequences for the abundance matching technique as well as for most semi-analytic models of galaxy formation. We provide analytic fitting formulae, derived from simulations that reproduce the observed baryon fractions, to correct halo masses and mass functions from DM-only simulations. The effect of baryon physics (AGN feedback in particular) on cluster number counts is about as large as changing the cosmology from Wilkinson Microwave Anisotropy Probe 7 to Planck, even when a moderately high-mass limit of M500 ≈ 1014 M⊙ is adopted. Thus, for precision cosmology the effects of baryons must be accounted for.