  Ultracold molecules offer brand new opportunities to probe the variation of
fundamental constants with unprecedented sensitivity. This paper summarizes
theoretical background and current constraints on the variation of fine
structure constant and electron-to-proton mass ratio, as well as proposals and
experimental efforts to measure the variations based on ultracold molecules. In
particular, we describe two novel spectroscopic schemes on ultracold molecules
which have greatly enhanced sensitivity to fundamental constants: resonant
scattering near Feshbach resonances and spectroscopy on close-lying energy
levels of ultracold molecules.
