  We present a comprehensive analysis of atomic hydrogen (HI) properties using
a semi-analytical model of galaxy formation and N-body simulations covering a
large cosmological volume at high resolution. We examine the HI mass function
and the HI density, characterizing both their redshift evolution and their
dependence on hosting halo mass. We analyze the HI 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 HI maps with HOD techniques. We
investigate the HI clustering properties relevant for future $21$~cm Intensity
Mapping (IM) experiments, including the HI bias and the shot noise level. The
HI bias increases with redshift and it is roughly flat on the largest scales
probed. The scale dependency 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 HI power spectrum and their
relation to the halo bias. We find that HI poor satellite galaxies play an
important role at the scales of the 1-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 SKA.
