A varying fine-structure constant?: evidence for and against Michael T. Murphy Institute of Astronomy, Univ. of Cambridge, UK Abstract Absorption lines in the spectra of distant quasars offer precise constraints on possible variations in the fundamental constants over cosmological time-scales and distances. We have recently proposed and demonstrated a new ``many-multiplet'' method of constraining changes in the fine structure constant, $\alpha\!\equiv\!e^2/\hbar c$, which allows an order of magnitude greater precision than previous methods. High resolution optical spectra -- all from the Keck/HIRES instrument -- of 143 absorption clouds over the redshift range $0.2 < z_{\rm abs} < 4.2$ yield statistical evidence for a smaller $\alpha$ at high redshift: $\da = (-0.57 \pm 0.11) \times 10^{-5}$. Known systematic effects cannot explain this surprisingly stubborn result, though strange isotopic abundances of Mg in the absorption clouds might explain some lower redshift results. Recently, initial independent analyses by other groups using spectra from a different spectrograph -- VLT/UVES -- seem to disagree with our purely Keck/HIRES findings. I will focus on the current status and future of the optical varying-$\alpha$ results and discuss potentially far more stringent constraints on $\alpha^2g_{\rm p}$ and $\alpha^2g_{\rm p}m_{\rm e}/m_{\rm p}$ from comparison of radio, millimetre and optical spectra.