  Damped Lyman alpha systems seen in the spectra of high-z QSOs arise in
high-density neutral gas in which molecular hydrogen (H_2) should be
conspicuous. Systematic searches to detect the H_2 lines redshifted into the
Lyman alpha forest at <3400\AA are now possible thanks to the unique
capabilities of UVES on the VLT. Here we summarise the present status of our on
going programme to search for H_2 in DLAs, discuss the physical conditions in
the systems where H_2 is detected and the implications of non-detections.
