  By measuring the fundamental constants in astrophysical objects one can test
basic physical principles as space-time invariance of physical laws along with
probing the applicability limits of the standard model of particle physics. The
latest constraints on the fine structure constant alpha and the
electron-to-proton mass ratio mu obtained from observations at high redshifts
and in the Milky Way disk are reviewed. In optical range, the most accurate
measurements have already reached the sensitivity limit of available
instruments, and further improvements will be possible only with next
generation of telescopes and receivers. New methods of the wavelength
calibration should be realized to control systematic errors at the sub-pixel
level. In radio sector, the main tasks are the search for galactic and
extragalactic objects suitable for precise molecular spectroscopy as well as
high resolution laboratory measurements of molecular lines to provide accurate
frequency standards. The expected progress in the optical and radio
astrophysical observations is quantified.
