Absorption lines of molecular hydrogen provide a most sensitive and powerful tool to understand the early evolution of galaxies and the intergalactic medium (IGM). In addition these lines are useful to investigate the space and time variations of the ratio of electron to proton mass. In this presentation I review the status of the search for H$_2$ at high $z$. High resolution spectrograph on ELTs will allow us to detect H$_2$ in the Lyman Break Galaxies (LBGs) and CO and HD in DLAs that already show H$_2$. This will allow us to investigate the astrochemistry at early epochs and study the physics of interstellar medium (ISM) at high redshifts.