The status of searches for possible variation in the constants of nature from astronomical observation of molecules is reviewed, focusing on the dimensionless constant representing the proton-electron mass ratio $mu=m_p/m_e$. The optical detection of H$_2$ and CO molecules with large ground-based telescopes (as the ESO-VLT and the Keck telescopes), as well as the detection of H$_2$ with the Cosmic Origins Spectrograph aboard the Hubble Space Telescope is discussed in the context of varying constants, and in connection to different theoretical scenarios. Radio astronomy provides an alternative search strategy bearing the advantage that molecules as NH$_3$ (ammonia) and CH$_3$OH (methanol) can be used, which are much more sensitive to a varying $mu$ than diatomic molecules. Current constraints are $|Deltamu/mu| < 5 times 10^{-6}$ for redshift $z=2.0-4.2$, corresponding to look-back times of 10-12.5 Gyrs, and $|Deltamu/mu| < 1.5 times 10^{-7}$ for $z=0.88$, corresponding to half the age of the Universe (both at 3$sigma$ statistical significance). Existing bottlenecks and prospects for future improvement with novel instrumentation are discussed.
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