This paper is organized in two parts. The first part (Sections 2-5) is
dedicated to the theory at T=0 and contains a pedagogical review of some
fundamental aspects related with the chiral symmetries of QCD, the U(1) problem
and its solution proposed by 'tHooft, Witten and Veneziano. In the second part
(Sections 6-14) we discuss the role of the U(1) axial symmetry for the phase
structure of QCD at finite temperature. One expects that, above a certain
critical temperature, also the U(1) axial symmetry will be 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, we
analyse a scenario in which the U(1) axial symmetry is still broken above the
chiral transition. We will show that this scenario can be consistently
reproduced in the full respect of the relevant QCD Ward Identities and also
using an effective Lagrangian model. A new order parameter is introduced for
the U(1) axial symmetry.Comment: Expanded version of a talk given at the ``Workshop on Quark-Gluon
Plasma and Relativistic Heavy Ions'', Frascati (Italy), January 14th-18th,
2002 (QGP2002); 61 pages, LaTeX file, + 6 PS figure
