  Microwave and submillimeter molecular transition frequencies between nearly
degenerated rotational levels, tunneling transitions, and mixed
tunneling-rotational transitions show an extremely high sensitivity to the
values of the fine-structure constant, alpha, and the electron-to-proton mass
ratio, mu. This review summarizes the theoretical background on
quantum-mechanical calculations of the sensitivity coefficients of such
transitions to tiny changes in alpha and mu for a number of molecules which are
usually observed in Galactic and extragalactic sources, and discusses the
possibility of testing the space- and time-invariance of fundamental constants
through comparison between precise laboratory measurements of the molecular
rest frequencies and their astronomical counterparts. In particular, diatomic
radicals CH, OH, NH+, and a linear polyatomic radical C3H in Pi electronic
ground state, polyatomic molecules NH3, ND3, NH2D, NHD2, H2O2, H3O+, CH3OH, and
CH3NH2 in their tunneling and tunneling-rotational modes are considered. It is
shown that sensitivity coefficients strongly depend on the quantum numbers of
the corresponding transitions. This can be used for astrophysical tests of
Einstein's Equivalence Principle all over the Universe at an unprecedented
level of sensitivity of ~10^-9, which is a limit three to two orders of
magnitude lower as compared to the current constraints on cosmological
variations of alpha and mu: Delta alpha/alpha < 10^-6, Delta mu/mu < 10^-7.
