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Absorption lines from molecular hydrogen ($\rm H_2$) in the spectra ofbackground sources are a powerful probe of the physical conditions inintervening cold neutral medium. At high redshift, $z>2$, $\rm H_2$ lines areconveniently shifted in the optical domain, allowing the use of ground-basedtelescopes to perform high-resolution spectroscopy, which is essential for aproper analysis of the cold gas. We describe recent observational progress,based on the development of efficient pre-selection techniques inlow-resolution spectroscopic surveys such as the Sloan Digital Sky Survey(SDSS). The next generation of spectrographs with high blue-throughput, such asCUBES, will certainly significantly boost the efficiency and outcome offollow-up observations. In this paper, we discuss high priority science casesfor CUBES, building on recent $\rm H_2$ observations at high-z: probing thephysical conditions in the cold phase of regular galaxies and outflowing gasfrom active galactic nucleus.

Molecular hydrogen in absorption at high redshifts. Science cases for CUBES