The status of varying constants: a review of the physics, searches and
  implications
  The observational evidence for the recent acceleration of the universe
demonstrates that canonical theories of cosmology and particle physics are
incomplete---if not incorrect---and that new physics is out there, waiting to
be discovered. A key task for the next generation of laboratory and
astrophysical facilities is to search for, identify and ultimately characterize
this new physics. Here we highlight recent developments in tests of the
stability of nature's fundamental couplings, which provide a direct handle on
new physics: a detection of variations will be revolutionary, but even improved
null results provide competitive constraints on a range of cosmological and
particle physics paradigms. A joint analysis of all currently available data
shows a preference for variations of $\alpha$ and $\mu$ at about the two-sigma
level, but inconsistencies between different sub-sets (likely due to hidden
systematics) suggest that these statistical preferences need to be taken with
caution. On the other hand, these measurements strongly constrain Weak
Equivalence Principle violations. Plans and forecasts for forthcoming studies
with facilities such as ALMA, ESPRESSO and the ELT, which should clarify these
issues, are also discussed, and synergies with other probes are briefly
highlighted. The goal is to show how a new generation of precision consistency
tests of the standard paradigm will soon become possible.
