We have searched for molecular hydrogen in damped Lyman-alpha (DLA) and sub-DLA systems at z>1.8 using UVES at the VLT. Out of the 33 systems in our sample, 8 have firm and 2 have tentative detections of associated H2 absorption lines. Considering that 3 detections were already known from past searches, H2 is detected in 13 to 20 percent of the newly-surveyed systems. We report new detections of molecular hydrogen at z=2.087 and 2.595 toward, respectively, Q 1444+014 and Q 0405-443, and also reanalyse the system at z=3.025 toward Q 0347-383. We find that there is a correlation between metallicity and depletion factor in both our sample and also the global population of DLA systems (60 systems in total). The DLA and sub-DLA systems where H2 is detected are usually amongst those having the highest metallicities and the largest depletion factors. Moreover, the individual components where H2 is detected have depletion factors systematically larger than other components in the profiles. In two different systems, one of the H2-detected components even has [Zn/Fe]>=1.4. These are the largest depletion factors ever seen in DLA systems. All this clearly demonstrates the presence of dust in a large fraction of the DLA systems. The mean H2 molecular fraction is generally small in DLA systems and similar to what is observed in the Magellanic Clouds. From 58 to 75 percent of the DLA systems have log f<-6. This can be explained if the formation rate of H2 onto dust grains is reduced in those systems, probably because the gas is warm (T>1000 K) and/or the ionizing flux is enhanced relative to what is observed in our Galaxy.
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