  Using the GAEA semi-analytic model, we analyse the connection between Damped
Lyman-$\alpha$ systems (DLAs) and HI in galaxies. Our state-of-the-art
semi-analytic model is tuned to reproduce the local galaxy HI mass function,
and that also reproduces other important galaxy properties, including the
galaxy mass - gas metallicity relation. To produce catalogs of simulated DLAs
we throw $10^5$ random lines of sight in a composite simulated volume: dark
matter haloes with log$(\frac{M_{200}}{ M_{\odot}}) \geq 11.5$ are extracted
from the Millennium Simulation, while for $9.2 \leq \log(\frac{M_{200}}{
M_{\odot}})<11.5$ we use the Millennium II, and for $8 \leq
\log(\frac{M_{200}}{M_{\odot}}) < 9.2$ a halo occupation distribution model. At
$2 < z < 3$, where observational data are more accurate, our fiducial model
predicts the correct shape of the column density distribution function, but its
normalization falls short of the observations, with the discrepancy increasing
at higher redshift. The agreement with observations is significantly improved
increasing both the HI masses and the disk radii of model galaxies by a factor
2, as implemented 'a posteriori' in our $2M-2R$ model. In the redshift range of
interest, haloes with $M_{200} \geq {10}^{11} M_{\odot}$ give the major
contribution to $\Omega_{\rm DLA}$, and the typical DLA host halo mass is $\sim
{10}^{11} M _{\odot}$. The simulated DLA metallicity distribution is in
relatively good agreement with observations, but our model predicts an excess
of DLAs at low metallicities. Our results suggest possible improvements for the
adopted modelling of the filtering mass and metal ejection in low-mass haloes.
