Do the fundamental constants of nature actually vary over cosmological time and distance scales? This question goes into the heart of cosmology, physics and particle physics. Several theories beyond the Standard Model of particle physics allow for varying of fundamental physical constants as well as dark energy if due to quintessence. Absorption lines recorded in the spectra of distant QSOs are "barcodes" of atomic structure and can effectively probe in space and in time the variability of the dimensionless constants such as the fine structure constant and proton-to-electron mass ratio which are related to fundamental forces of nature. The current observational status will be critically reviewed trying to understand the controversial results obtained. The prospects of the ESO Large Program just taking off and of the ESPRESSO and CODEX projects will be outlined. These will make possible significant improvements and opening an entirely new field of research in case of positive results. I will also present radio observations of dense pre-stellar molecular clouds in the Milky Way. These show an intriguing velocity offsets between the NH3 inversion transition and other molecules which does not have a simple explanation. Taken at face value the measurements could imply spatial variations of the mp/me ratio of about 30 ppb (parts-per-billion). However, these results are in conflict with atomic clock experiments in the solar system by several orders of magnitude, thus requiring a sort of chameleon-type scalar fields.