  The observational evidence for the recent acceleration of the universe shows
that canonical theories of cosmology and particle physics are incomplete and
that new physics is out there, waiting to be discovered. A compelling task for
astrophysical facilities is to search for, identify and ultimately characterize
this new physics. I present very recent developments in tests of the stability
of nature's fundamental constants, as well as their impact on physics paradigms
beyond the standard model. Specifically I discuss new observational constraints
at low redshifts and at the BBN epoch, and highlight their different
implications for canonical quintessence-type models and for non-canonical
string-theory inspired models. Finally I also present new forecasts, based on
realistic simulated data, of the gains in sensitivity for these constraints
expected from ELT-HIRES, on its own and in combination with Euclid.
