  We investigate how well the redshift distribution of a population of
extragalactic objects can be reconstructed using angular cross-correlations
with a sample whose redshifts are known. We derive the minimum variance
quadratic estimator, which has simple analytic representations in very
applicable limits and is significantly more sensitive than earlier proposed
estimation procedures. This estimator is straightforward to apply to
observations, it robustly finds the likelihood maximum, and it conveniently
selects angular scales at which fluctuations are well approximated as
independent between redshift bins and at which linear theory applies. We find
that the linear bias times number of objects in a redshift bin generally can be
constrained with cross-correlations to fractional error (10^2 n/N)^1/2, where N
is the total number of spectra per dz and n is the number of redshift bins
spanned by the bulk of the unknown population. The error is often independent
of the sky area and sampling fraction. Furthermore, we find that sub-percent
measurements of the angular source density per unit redshift, dN/dz, are in
principle possible, although cosmic magnification needs to be accounted for at
fractional errors of <~ 10 per cent. We discuss how the sensitivity to dN/dz
changes as a function of photometric and spectroscopic depth and how to
optimize the survey strategy to constrain dN/dz. We also quantify how well
cross-correlations of photometric redshift bins can be used to self-calibrate a
photometric redshift sample. Simple formulae that can be quickly applied to
gauge the utility of cross correlating different samples are given.
