We have searched for molecular hydrogen in damped Lyman- \alpha ( DLA ) and sub-DLA systems at high redshift ( z _ { abs } > 1.8 ) using UVES at the VLT down to a detection limit of typically N ( H _ { 2 } ) = 2 \times 10 ^ { 14 } cm ^ { -2 } .
Out of the 33 systems in our sample , 8 have firm and 2 have tentative detections of associated H _ { 2 } absorption lines .
Considering that 3 detections were already known from past searches , molecular hydrogen is detected in 13 to 20 percent of the newly-surveyed systems .
We report new detections of molecular hydrogen at z _ { abs } = 2.087 and 2.595 toward , respectively , Q 1444 + 014 and Q 0405 - 443 , and also reanalyse the system at z _ { abs } = 3.025 toward Q 0347 - 383 .

In all of the systems , we measure metallicities relative to Solar , [ X/H ] ( with either X = Zn , or S , or Si ) , and depletion factors of Fe , [ X/Fe ] , supposedly onto dust grains , and compare the characteristics of our sample with those of the global population of DLA systems ( 60 systems in total ) .
We find that there is a correlation between metallicity and depletion factor in both our sample and also the global population of DLA systems .
Although H _ { 2 } molecules are detected in systems with [ Zn/Fe ] as small as 0.3 , the DLA and sub-DLA systems where H _ { 2 } is detected are usually amongst those having the highest metallicities and the largest depletion factors .
In particular , H _ { 2 } is detected in the five systems having the largest depletion factors .
Moreover , the individual components where H _ { 2 } is detected have depletion factors systematically larger than other components in the profiles .
In two different systems , one of the H _ { 2 } -detected components even has [ Zn/Fe ] \geq 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 H _ { 2 } molecular fraction , f = 2 N ( H _ { 2 } ) / [ 2 N ( H _ { 2 } ) + N ( H i ) ] , is generally small in DLA systems ( typically \log f < -1 ) and similar to what is observed in the Magellanic Clouds .
There is no correlation between the observed amount of H _ { 2 } and the H i column density .
In fact , two systems where H _ { 2 } is detected have \log N ( H i ) < 20.3 and , therefore , are sub-DLA systems .
From 58 to 75 percent of the DLA systems have \log f < -6 .
This can be explained if the formation rate of H _ { 2 } 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 .