We investigate the structural, kinematical and chemical properties of globular cluster systems (GCSs) in galaxies of different Hubble types in a self-consistent manner based on high-resolution cosmological N-body simulations combined with semi-analytic models of galaxy and globular cluster (GC) formation. We focus on correlations between the physical properties of GCSs and those of their host galaxies for ∼105 simulated galaxies located at the centres of dark matter haloes (i.e. we do not consider satellite galaxies in subhaloes). Our principal results, which can be tested against observations, are as follows. The majority (∼90 per cent) of GCs currently in haloes are formed in low-mass galaxies at redshifts greater than 3 with mean formation redshifts of z= 5.7 (12.7 Gyr ago) and 4.3 (12.3 Gyr ago) for metal-poor GCs (MPCs) and metal-rich GCs (MRCs), respectively. About 52 per cent of galaxies with GCs show clear bimodality in their metallicity distribution functions, though less luminous galaxies with MB fainter than −17 are much less likely to show bimodality owing to little or no MRCs. The number fraction of MRCs does not depend on Hubble type but is generally smaller for less luminous galaxies. The specific frequencies (SN) of GCSs are typically higher in ellipticals (SN∼ 4.0) than in spirals (SN∼ 1.8), and higher again (SN∼ 5.0) for galaxies located at the centres of clusters of galaxies. The total number of GCs per unit halo mass does not depend strongly on MB or Hubble type of the host galaxy. The mean metallicities of MPCs and MRCs depend on MB such that they are higher in more luminous galaxies, though the dependence is significantly weaker for MPCs. The spatial distributions of MRCs are more compact than those of MPCs and we find that the half-number radii of MPCs (re,mpc) correlate with the halo masses (Mh) such that re,mpc∝Mh0.18. There is no significant difference in velocity dispersions between MPCs and MRCs. We qualitatively compare our results to observational data where possible. Finally, we discuss these results in the wider context of galaxy formation and evolution.