  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.
