Recent variations of fundamental parameters and their implications for gravitation

We compare the sensitivity of a recent bound on time variation of the fine structure constant from optical clocks with bounds on time varying fundamental constants from atomic clocks sensitive to the electron-to-proton mass ratio, from radioactive decay rates in meteorites, and from the Oklo natural reactor. Tests of the Weak Equivalence Principle also lead to comparable bounds on present time variations of constants, as well as putting the strongest limits on variations tracking the gravitational potential. For recent time variations, the "winner in sensitivity" depends on possible relations between the variations of different couplings in the standard model of particle physics. WEP tests are currently the most sensitive within scenarios with unification of gauge interactions. A detection of time variation in atomic clocks would favour dynamical dark energy and put strong constraints on the dynamics of a cosmological scalar field.