  We present a short review of the current quasar (QSO) absorption line
constraints on possible variation of the fine structure constant, alpha =
e^2/(hbar*c). Particular attention is paid to recent optical Keck/HIRES spectra
of 49 absorption systems which indicate a smaller alpha in the past (Murphy et
al. 2001, Webb et al. 2001). Here we present new preliminary results from 128
absorption systems: da/a = (-0.57 +/- 0.10) x 10^{-5} over the redshift range
0.2 < z < 3.7, in agreement with the previous results. Known potential
systematic errors cannot explain these results. We compare them with strong
`local' constraints and discuss other (radio and millimetre-wave) QSO
absorption line constraints on variations in alpha^2 * g_p and alpha^2 * g_p *
m_e/m_p (g_p is the proton g-factor and m_e/m_p is the electron/proton mass
ratio). Finally, we discuss future efforts to rule out or confirm the current
5.7 sigma optical detection.
