  We review some string-inspired theoretical models which incorporate a
correlated spacetime 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
unnatural fine-tunings 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. Recent claims by
Bekenstein that fine-structure-constant variability does not imply detectable
violations of the equivalence principle are shown to be untenable.
