Astronomical and laboratory searches for space-time variation of fundamental constants

We review recent activity searching for variations in the fundamental constants of nature in quasar absorption spectra and in the laboratory. While research in this direction has been ongoing for many decades, the topic has recently been stimulated by astronomical evidence for spatial variation in the fine-structure constant, alpha. This result could be confirmed using different quasar data and atomic clock measurements, but there are significant challenges to obtain the required accuracy. We review existing measurements and discuss some of the most promising systems where any variations would be strongly enhanced.