  We discuss the role of the U(1) axial symmetry for the phase structure of QCD
at finite temperature. We expect that, above a certain critical temperature,
also the U(1) axial symmetry will be (effectively) restored. We will try to see
if this transition has (or has not) anything to do with the usual chiral
transition: various possible scenarios are discussed. In particular, supported
by recent lattice results, we analyse a scenario in which a U(1)-breaking
condensate survives across the chiral transition. This scenario can be
consistently reproduced using an effective Lagrangian model. The effects of the
U(1) chiral condensate on the slope of the topological susceptibility in the
full theory with quarks are studied: we find that this quantity (in the chiral
limit of zero quark masses) acts as an order parameter for the U(1) axial
symmetry above the chiral transition. Further information on the new U(1)
chiral order parameter is derived from the study (at zero temperature) of the
radiative decays of the pseudoscalar mesons in two photons: a comparison of our
results with the experimental data is performed.
