String theory, cosmology and varying constants

In string theory the coupling `constants' appearing in the low-energy effective Lagrangian are determined by the vacuum expectation values of some (a priori) mass less scalar fields (dilaton, moduli). This naturally leads one to expect a correlated variation of all the coupling constants, and an associated violation of the equivalence principle. We review some string-inspired theoretical models which incorporate such a space time variation of coupling constants while remaining naturally compatible both with phenomenological constraints coming from geochemical data (Oklo; Rhenium decay) and with present equivalence principle tests. Barring a very unnatural fine-tuning of parameters, a variation of the fine-structure constant as large as that recently `observed' by Webb et al. in quasar absorption spectra appears to be incompatible with these phenomenological constraints. Independently of any model, it is emphasized that the best experimental probe of varying constants are high-precision tests of the universality of free fall, such as MICROSCOPE and STEP. This revised version was published online in July 2006 with corrections to the Cover Date.