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The observational evidence for the recent acceleration of the universe showsthat canonical theories of cosmology and particle physics are incomplete andthat new physics is out there, waiting to be discovered. A compelling task forastrophysical facilities is to search for, identify and ultimately characterizethis new physics. I present very recent developments in tests of the stabilityof nature's fundamental constants, as well as their impact on physics paradigmsbeyond the standard model. Specifically I discuss new observational constraintsat low redshifts and at the BBN epoch, and highlight their differentimplications for canonical quintessence-type models and for non-canonicalstring-theory inspired models. Finally I also present new forecasts, based onrealistic simulated data, of the gains in sensitivity for these constraintsexpected from ELT-HIRES, on its own and in combination with Euclid.

Varying fundamental constants and dark energy in the ESPRESSO era