  Invoking the Peccei-Quinn (PQ) solution to the strong CP problem substitutes
the puzzle of why $\theta_{qcd}$ is so small with the puzzle of why the PQ
symmetry is of such high quality. Cosmological and astrophysical considerations
raise further puzzles. This paper explores this issues in several contexts:
string theory and field theory, and theories without and with low energy
supersymmetry. Among the questions studied are whether requiring axion dark
matter can account for the quality of the PQ symmetry, to which the answer is
sometimes yes. In non-supersymmetric theories, we find $f_a = 10^{12}$ GeV is
quite plausible. In gauge mediation, cosmological constraints on pseudomoduli
place $f_a$ in this range, and require that the gravitino mass be of order an
MeV.
