  In these two lectures given at the 1997 Zakopane workshop on "New
Developments in Quantum Field Theory" we review recent results on universal
fluctuations in QCD Dirac spectra. We start the first lecture with a review of
some general properties of Dirac spectra. It will be argued that there is an
intimate relation between chiral symmetry breaking and correlations of Dirac
eigenvalues. In particular, we will focus on the microscopic spectral density
density, i.e. the spectral density near zero virtuality on the scale of a
typical level spacing. The relation with Leutwyler-Smilga sum-rules will be
discussed. Standard methods for the statistical analysis of quantum spectra
will be reviewed. Recent results on the application of Random Matrix Theory to
spectra of 'complex' systems will be summarized. This leads to the introduction
of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD
partition function. In the second lecture the chiral random matrix model will
be compared to QCD and some of its properties will be discussed. Our central
conjecture is that correlations of QCD Dirac spectra are described by chRMT. We
will review recent results showing that the microscopic spectral density and
eigenvalue correlations near zero virtuality are strongly universal. Lattice
QCD results for the microscopic spectral density and for correlations in the
bulk of the spectrum will be presented. In all cases that have been considered,
the correlations are in perfect agreement with chRMT. We will end the second
lecture with a review of chiral Random Matrix Theory at nonzero chemical
potential. New features of spectral universality in nonhermitean matrices will
be discussed.
