Problems of cosmological variability of fundamental physical constants

We discuss recent laboratory experiments and astronomical observations aime d at testing the possible space-time variability of fundamental physical co nstants, predicted by the modern theory Specifically, we consider two of th e dimensionless physical parameters which are important for atomic and mole cular physics: the fine-structure constant alpha and the electron-to-proton mass ratio mu. We review the current status of such experiments and critic ally analyze recent claims of a detection of the variability of the fine-st ructure constant on the cosmological time scale. We stress that such a dete ction remains to be checked by future experiments and observations. The tig htest of the firmly established upper limits, derived from analyses of quas ar spectra formed at the early cosmological epoch (similar to 10(10) yr ago ), read: \(alpha) over dot/alpha \ < 1.1 x 10(-14) yr(-1), \(mu) over dot/m u \ < 1.5 x 10(-14) yr(-1). These limits may be used as an effective tool f or selection of theoretical models which predict space-time variations of p hysical constants.