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Using the GAEA semi-analytic model, we analyse the connection between Damped Lyman-$α$ 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 $Ω_{\rm DLA}$, 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.