  The ionization history of the universe provides a major source of ambiguity
in constraining cosmological models using small angular scale microwave
background anisotropies. To clarify these issues, we consider a complete
treatment of Compton scattering to second order, an approach which may be
applicable to other astrophysical situations. We find that only the ${\cal O}
(v)$ Doppler effect, and the ${\cal O} (v\delta)$ Vishniac effect are important
for recent last scattering epochs; the ${\cal O} (v^2)$ Doppler effect is not
significant on any angular scale, and other higher-order effects are completely
negligible. However the ${\cal O}(v^2)$ effect does lead to \Cy\ distortions,
which although generally below current constraints, set an unavoidable minimum
level in reionization models. We consider the small-angle approximation for the
Vishniac effect in detail, and show several improvements over previous
treatments, particularly for low $\Omega_0$. For standard cold dark matter
models, the effect of reionization is to redistribute the anisotropies to
arcminute scales; late reionization leads to partially erased primary
fluctuations and a secondary contribution of comparable magnitude. Using recent
anisotropy limits from the ATCA experiment, we set new constraints on baryonic
dark matter models. Stronger constraints are imposed (in second order) upon
models with higher Hubble constant, steeper $n$, and higher density. These
limits depend on the specific ionization history assumed, but the factor gained
by lowering the ionization fraction is generally small, and may be tested by
currently-planned
