  In the first part of these lectures we discuss the infrared limit of the
spectrum of the QCD Dirac operator. We discuss the global symmetries of the QCD
partition function and show that the Dirac spectrum near zero virtuality is
determined by the pattern of spontaneous chiral symmetry breaking of a QCD-like
partition function with additional bosonic valence quarks and their
super-symmetric partners. We show the existence of an energy scale below which
the fluctuations of the QCD Dirac spectrum are given by a chiral Random Matrix
Theory (chRMT) with the global symmetries of the QCD partition function.
Physically, for valence quark masses below this scale the partition function is
dominated by the zero momentum modes. In the theory of disordered systems, this
energy scale is known as the Thouless energy. In the second part of these
lectures we discuss chRMT as a schematic model for the QCD partition function
at nonzero temperature and chemical potential. We discuss novel features
resulting from the non-Hermiticity of the Dirac operator. The analysis by
Stephanov of the failure of the quenched approximation, the phase diagram of
the chRMT partition function, as well as the properties of Yang-Lee zeros are
discussed. We argue that a localization transition does not occur in the
presence of light quarks. Several results will be derived in full detail. We
mention the flavor symmetries of the QCD Dirac operator for two colors, the
calculation of the valence quark mass dependence of the chiral condensate and
the reduction of the chRMT partition function to the finite volume partition
function.
