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