Astrophysical studies of redshifted spectral lines provide a powerful probe of putative changes in fundamental constants across a large lookback time. After reviewing the current status of this field, I will describe a new technique to measure such changes, using multiple radio OH lines. This allows a simultaneous measurement of changes in the fine structure constant alpha, the proton g-factor g_p and the electron-proton mass ratio m_e/m_p, using OH lines from a single cosmologically distant object. I will also present the first detections of the 18cm OH satellite lines at cosmological distances, from the z~0.25 source PKS 1413+135 and the z~0.765 gravitational lens towards PMN J0134-0931. The 1720 MHz and 1620 MHz profiles are conjugate in both cases,with the sum of their optical depths consistent with noise; this implies that the lines arise in precisely the same gas. A comparison between the line redshifts can thus be used to test for any changes in the above constants, over a large fraction of the age of the Universe.