ABSTRACT We present a comprehensive analysis of atomic hydrogen (H i) properties using a semi-analytical model of galaxy formation and N-body simulations covering a large cosmological volume at high resolution. We examine the H i mass function and the H i density, characterizing both their redshift evolution and their dependence on hosting halo mass. We analyse the H i content of dark matter haloes in the local Universe and up to redshift z = 5, discussing the contribution of different galaxy properties. We find that different assembly history plays a crucial role in the scatter of this relation. We propose new fitting functions useful for constructing mock H i maps with halo occupation distribution techniques. We investigate the H i clustering properties relevant for future 21 cm intensity mapping (IM) experiments, including the H i bias and the shot-noise level. The H i bias increases with redshift and it is roughly flat on the largest scales probed. The scale dependence is found at progressively larger scales with increasing redshift, apart from a dip feature at z = 0. The shot-noise values are consistent with the ones inferred by independent studies, confirming that shot noise will not be a limiting factor for IM experiments. We detail the contribution from various galaxy properties on the H i power spectrum and their relation to the halo bias. We find that H i poor satellite galaxies play an important role at the scales of the one-halo term. Finally, we present the 21 cm signal in redshift space, a fundamental prediction to be tested against data from future radio telescopes such as Square Kilometre Array.