We present the results of a search for cold gas at high redshift along quasar lines of sight carried out without any a priori assumption on the neutral atomic-hydrogen content of the absorption-line systems. To do this, we systematically looked for neutral-carbon (C i) λλ1560, 1656 transition lines in 41 696 low-resolution quasar spectra (1.5 < z em < 4.46) from the SDSS-II -Data Release 7 -database. C i absorption lines should indeed probe the shielded gas in the neutral interstellar medium of galaxies more efficiently than traditional tracers such as neutral atomic-hydrogen (H i) damped Lyman-α (DLA) and/or Mg ii systems. We built up a sample of 66 C i absorbers with redshifts in the range 1.5 < z < 3.1 and rest-frame equivalent widths 0.1 < W r (λ1560) < 1.7 Å. The completeness limit of our survey is W r,lim (λ1560) 0.4 Å. C i systems stronger than that are more than one hundred times rarer than DLAs at z abs = 2.5. The number of C i systems per unit redshift is found to increase significantly below z = 2. We suggest that these C i absorbers are closely related to the process of star formation and the production of dust in galaxies and that their cosmic evolution is driven by the interplay between dust shielding and the evolution of the ultra-violet background at ∼10 eV. We derive the neutral atomic-hydrogen content of the C i systems observable from the southern hemisphere from VLT/UVES spectroscopy and find that a majority of them are sub-DLAs with N(H i) ∼ 10 20 atoms cm −2 . The dust content of these absorbers is yet significant as seen from the redder optical colours of the corresponding background quasars and their reddened spectral energy distributions, with E(B − V) values up to ∼0.3. The overall N(H i) distribution of C i systems is, however, relatively flat. As a consequence, among the C i systems classifying as DLAs, there is a probable excess of strong DLAs with log N(H i) > 21 (atoms cm −2 ) compared to systematic DLA surveys. Whilst the dust content of such systems is significant, their dust-to-gas ratio must still be limited. Indeed, strong DLAs having large amounts of shielded gas and dust producing stronger reddening and extinction of the background quasars, if they exist, should have been missed in the current magnitude-limited quasar sample. We study the empirical relations between W r (C i), N(H i), E(B − V), and the strength of the 2175 Å extinction feature. The latter is detected in about 30% of the C i absorbers. We show that, at equal amount of reddening, the 2175 Å feature is weak compared to Galactic lines of sight. This is probably the consequence of current or past star formation in the vicinity of the C i systems. We also find that the strongest C i systems tend to have the largest amounts of dust and that the metallicity of the gas and its molecular fraction is likely to be high in a large number of cases. The C i-absorber sample presented here hence provides ideal targets for detailed studies of the dust composition and molecular species at high redshift.