  We discuss conditions that should be satisfied by axion models for solving
the strong CP problem. It has been observed that Planck scale effects may
render the axion models ineffective if there are gauge invariant operators of
dimension less than 10 which break explicitly the Peccei-Quinn (PQ) symmetry.
We argue that only those operators formed of fields which have vacuum
expectation values are dangerous. Supersymmetric axion models fail to prevent
even this restricted class of operators. Furthermore, the models that relate
the PQ scale and the supersymmetry breaking scale are particularly sensitive to
gauge invariant PQ-breaking operators. By contrast, in non-supersymmetric
composite axion models the PQ scale arises naturally, and the dangerous
operators can be avoided. However, the composite axion models contain heavy
stable particles which are cosmologically ruled out. Another problem is a
Landau pole for the QCD coupling constant. Both these problems may be solved if
the unification of color with the gauge interactions which bind the axion could
be achieved.
