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An overview is presented of the H$_2$ quasar absorption method to search fora possible variation of the proton--electron mass ratio $\mu=m_p/m_e$ on acosmological time scale. Details of the analysis of astronomical spectra,obtained with large 8--10 m class optical telescopes, equipped withhigh-resolution echelle grating based spectrographs, are explained. The methodsand results of the laboratory molecular spectroscopy of H$_2$, in particularthe laser-based metrology studies for the determination of rest wavelengths ofthe Lyman and Werner band absorption lines, are reviewed. Theoretical physicsscenarios delivering a rationale for a varying $\mu$ will be discussed briefly,as well as alternative spectroscopic approaches to probe variation of $\mu$,other than the H$_2$ method. Also a recent approach to detect a dependence ofthe proton-to-electron mass ratio on environmental conditions, such as thepresence of strong gravitational fields, will be highlighted. Currently some 56H$_2$ absorption systems are known and listed. Their usefulness to detect$\mu$-variation is discussed, in terms of column densities and brightness ofbackground quasar sources, along with future observational strategies. Theastronomical observations of ten quasar systems analyzed so far set aconstraint on a varying proton-electron mass ratio of $|\Delta\mu/\mu| < 5\times 10^{-6}$ (3-$\sigma$), which is a null result, holding for redshifts inthe range $z=2.0-4.2$. This corresponds to look-back times of 10--12.4 billionyears into cosmic history. Attempts to interpret the results from these 10H$_2$ absorbers in terms of a spatial variation of $\mu$ are currently hamperedby the small sample size and their coincidental distribution in a relativelynarrow band across the sky.

Search for a drifting proton--electron mass ratio from H$_2$