  We determine the behaviour of a time-varying fine structure 'constant'
$\alpha (t)$ during the early and late phases of universes dominated by the
kinetic energy of changing $\alpha (t)$, radiation, dust, curvature, and
lambda, respectively. We show that after leaving an initial vacuum-dominated
phase during which $\alpha$ increases, $\alpha$ remains constant in universes
like our own during the radiation era, and then increases slowly, proportional
to a logarithm of cosmic time, during the dust era. If the universe becomes
dominated by negative curvature or a positive cosmological constant then
$\alpha$ tends rapidly to a constant value. The effect of an early period of de
Sitter or power-law inflation is to drive $\alpha$ to a constant value. Various
cosmological consequences of these results are discussed with reference to
recent observational studies of the value of $\alpha$ from quasar absorption
spectra and to the existence of life in expanding universes.
