  If axions exist they are efficiently produced in the hot and dense interior
of stars, providing a novel energy-loss mechanism. In order to avoid a conflict
with the observed properties of stars, one can derive a lower limit on the
Peccei-Quinn scale (an upper limit on the axion mass). In the early universe,
axions are produced by the ``misalignment mechanism'' and the emission from
global strings as well as the relaxation of the string-domain wall system
formed at the QCD phase transition. In order to avoid an ``overclosed
universe'' the Peccei-Quinn scale must obey an upper limit (a lower limit on
the axion mass). The current values of these bounds are reviewed. There remains
a ``window of opportunity'' $10^{-5}\,\eV\lapprox m_a\lapprox 10^{-2}\,\eV$,
with large uncertainties on either side, where axions could still exist.
[Contribution to the XVth Moriond workshop, 21-28 Jan. 1995, Villars-sur-Ollon,
Switzerland.]
