  Spectroscopy of hydrogen can be used for a search into physics beyond the
Standard Model. Differences between the absorption spectra of H$_2$ as observed
at high redshift and those measured in the laboratory can be interpreted in
terms of possible variations of the proton-electron mass ratio. Investigation
of some ten of such absorbers in the redshift range $z= 2.0-4.2$ yields a
constraint of $|\Delta\mu/\mu|< 5 \times 10^{-6}$ at 3$\sigma$. Observation of
H$_2$ from the photospheres of white dwarf stars inside our Galaxy delivers a
constraint of similar magnitude on a dependence of $\mu$ on a gravitational
potential $10^4$ times as strong as on the Earth's surface. Laser-based
precision measurements of dissociation energies, vibrational splittings and
rotational level energies in H$_2$ molecules and their deuterated isotopomers
HD and D$_2$ produce values for the rovibrational binding energies fully
consistent with quantum ab initio calculations including relativistic and
quantum electrodynamical (QED) effects. Similarly, precision measurements of
high-overtone vibrational transitions of HD$^+$ ions, captured in ion traps and
sympathetically cooled to mK temperatures, also result in transition
frequencies fully consistent with calculations including QED corrections.
Precision measurements of inter-Rydberg transitions in H$_2$ can be
extrapolated to yield accurate values for level splittings in the H$_2^+$-ion.
These comprehensive results of laboratory precision measurements on neutral and
ionic hydrogen molecules can be interpreted to set bounds on the existence of
possible fifth forces and of higher dimensions, phenomena describing physics
beyond the Standard Model.
