We describe the results of a search for time variability of the fine structure constant alpha using absorption systems in the spectra of distant quasars. Three large optical data sets and two 21 cm and mm absorption systems provide four independent samples, spanning approximately 23% to 87% of the age of the universe. Each sample yields a smaller alpha in the past and the optical sample shows a 4 sigma deviation: Delta alpha/alpha = -0.72+/-0.18 x 10(-5) over the redshift range 0.5<z<3.5. We find no systematic effects which can explain our results. The only potentially significant systematic effects push Delta alpha/alpha towards positive values; i.e., our results would become more significant were we to correct for them.