The discovery of the first damped Lyman alpha (DLA) system in the early 1970s followed by the recognition that DLAs arise in intervening galaxies opened up a new field of galaxy evolution research. These highest HI column density absorption-line systems trace the bulk of the observed neutral gas in the Universe, and therefore, have been used as powerful probes of galaxy formation and evolution back to the redshifts of the most distant quasars. The history and progress of DLA research over the past several decades is reviewed here. Larger datasets and deeper surveys, particularly over the last couple of years, have improved our knowledge of the neutral gas content and distribution in the Universe at all observable redshifts, including the present epoch. New results on the statistics of DLAs at $z<1.65$ from our HST-UV surveys are presented and discussed in the context of recent results at $z=0$ and at high redshift. We find that $\Omega_{DLA}(z>0)$ remains roughly constant to within the uncertainties; the $z=0$ value of the neutral gas mass density, $\Omega_{g}$, is a factor of $\approx 2$ less than $\Omega_{DLA}$. The DLA incidence, $n(z)$, undergoes rapid evolution between redshifts 5 and 2, but is consistent with the no-evolution curve in the current concordance cosmology for $z\lesssim 2$. We also show that if the local Schmidt law relating surface density of gas and star formation rate (SFR) is valid at the DLA redshifts, then the DLA SFR density is too low for them to provide a significant contribution to the cosmic star formation history (SFH) at $z\gtrsim 1$. This implies that the DLAs are unlikely to be the same population as the star forming galaxies (i.e. the Lyman break and sub-millimetre galaxies) that dominate the SFH of the high redshift Universe. We suggest that this discrepancy and the DLA “missing metals” problem could be the result of missing very high column density gas due to its very small absorption cross-section.