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 waiting to be discovered. A key task for astrophysical facilities is to search for, identify and ultimately characterize this new physics. I will discuss recent developments in tests of the stability of nature's fundamental constants, which provide a direct handle on new physics. I will summarize the theoretical context and expectations, the latest observational constraints, and some implications for dark energy models. Time permitting I will also briefly highlight the gains in sensitivity expected from next-generation facilities.