  Recent theoretical ideas and observational claims suggest that the fine
structure constant alpha may be variable. We examine a spectrum of models in
which alpha is a function of a scalar field. Specifically, we consider three
scenarios: oscillating alpha, monotonic time variation of alpha, and
time-independent alpha that is spatially varying. We examine the constraints
imposed upon these theories by cosmological observations, particle detector
experiments, and ``fifth force'' experiments. These constraints are very strong
on models involving oscillation, but cannot compete with bounds from the Oklo
sub-nuclear reactor on models with monotonic time-like variation of alpha. One
particular model with spatial variation is consistent with all current
experimental and observational measurements, including those from two seemingly
conflicting measurements of the fine structure constant using the many
multiplet method on absorption lines.
