Details of how the primordial plasma recombined and how the universe later
reionized are currently somewhat uncertain. This uncertainty can restrict the
accuracy of cosmological parameter measurements from the Cosmic Microwave
Background (CMB). More positively, future CMB data can be used to constrain the
ionization history using observations. We first discuss how current
uncertainties in the recombination history impact parameter constraints, and
show how suitable parameterizations can be used to obtain unbiased parameter
estimates from future data. Some parameters can be constrained robustly,
however there is clear motivation to model recombination more accurately with
quantified errors. We then discuss constraints on the ionization fraction
binned in redshift during reionization. Perfect CMB polarization data could in
principle distinguish different histories that have the same optical depth. We
discuss how well the Planck satellite may be able to constrain the ionization
history, and show the currently very weak constraints from WMAP three-year
data.Comment: Changes to match MNRAS accepted versio