The idea of possible time or space variations of the 'fundamental' constants of nature, although not new, is only now beginning to be actively considered by large numbers of researchers in the particle physics, cosmology and astrophysics communities. This revival is mostly due to the claims of possible detection of such variations, in various different contexts and by several groups. I present the current theoretical motivations and expectations for such variations, review the current observational status and discuss the impact of a possible confirmation of these results in our views of cosmology and physics as a whole.