We summarize the attempts by our group and others to derive constraints on variations of fundamental constants over cosmic time using quasar absorption lines. Most upper limits reside in the range 0.5-1.5x10-5 at the 3sigma level over a redshift range of approximately 0.5-2.5 for the fine-structure constant, alpha, the proton-to-electron mass ratio, mu, and a combination of the proton gyromagnetic factor and the two previous constants, gp(alpha^2/mu)^nu, for only one claimed variation of alpha. It is therefore very important to perform new measurements to improve the sensitivity of the numerous methods to at least <0.1x10-5 which should be possible in the next few years. Future instrumentations on ELTs in the optical and/or ALMA, EVLA and SKA pathfinders in the radio will undoutedly boost this field by allowing to reach much better signal-to-noise ratios at higher spectral resolution and to perform measurements on molecules in the ISM of high redshift galaxies.
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