We study the nature of Damped Lyman -Alpha systems (DLAs) by means of a comparison between observed abundances and models of chemical evolution of galaxies of different morphological type. In particular, we compare for the first time the abundance ratios as functions of metallicity and redshift with dust-corrected data. We have developed detailed models following the evolution of several chemical elements (H, D, He, C, N, O, Ne, Mg, Si, S, Fe, Ni and Zn) for elliptical, spiral and irregular galaxies. Each of the models is calibrated to reproduce the main features of a massive elliptical, the Milky Way and the LMC, respectively. In addition, we run some models also for dwarf irregular starburst galaxies. All the models share the same uptodate nucleosynthesis prescriptions but differ in their star formation histories. The role of SNe of different type (II, Ia) is studied in each galaxy model together with detailed and up to date nucleosynthesis prescriptions. Our main conclusions are: 1) when dust depletion is taken into account most of the claimed alpha/Fe overabundances disappear and DLAs show solar or subsolar abundance ratios. 2) The majority of DLAs can be explained either by disks of spirals observed at large galactocentric distances or by irregular galaxies like the LMC or by starburst dwarf irregulars observed at different times after the last burst of star formation. 3) Elliptical galaxies cannot be DLA systems since they reach a too high metallicity at early times and their abundance ratios show overabundances of $α$-elements relative to Fe over a large range of [Fe/H]. 4) The observed neutral gas cosmic evolution is compared with our predictions but no firm conclusions can be drawn in the light of the available data. Comment: MNRAS, accepted for publication, 19 pages, 14 figs