  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
$|\Delta\mu/\mu| < 5 \times 10^{-6}$ for redshift $z=2.0-4.2$, corresponding to
look-back times of 10-12.5 Gyrs, and $|\Delta\mu/\mu| < 1.5 \times 10^{-7}$ 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.
