Particle-physics models with pseudo Nambu-Goldstone bosons (PNGB's) are characterized by two mass scales: a global spontaneous-symmetry-breaking scale f and a soft (explicit) symmetry-breaking scale LAMBDA. We investigate general model-insensitive constraints on this two-dimensional parameter space arising from the cosmological and astrophysical effects of PNGB's. In particular, we study constraints arising from vacuum misalignment and thermal production of PNGB's, topological defects, and the cosmological effects of PNGB decay products, as well as astrophysical constraints from stellar PNGB emission. Bounds on the Peccei-Quinn axion scale, 10(10) GeV less-than-or-similar-to f(PQ) less-than-or-similar-to 10(10)-10(12) GeV, emerge as a special case, where the soft breaking scale is fixed at LAMBDA(QCD) congruent-to 100 MeV.