  In these lectures we review recent results on universal fluctuations of QCD
Dirac spectra and applications of Random Matrix Theory (RMT) to QCD. We review
general properties of Dirac spectra and discuss the 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. The success of applications of
RMT to spectra of 'complex' systems leads us to the introduction of a chiral
Random Matrix Theory (chRMT) with the global symmetries of the QCD partition
function. Our central conjecture is that it decribes correlations of QCD Dirac
spectra. We will review recent universality proofs supporting this conjecture.
Lattice QCD results for the microscopic spectral density and for correlations
in the bulk of the spectrum are shown to be in perfect agreement with chRMT. We
close with a review of chRMT at nonzero chemical potential. Novel features of
spectral universality in nonhermitean matrices will be discussed. As an
illustration of mathematical methods used in RMT several important recent
results will be derived in all details. We mention the derivation of the
microscopic spectral density, the universality proof by Akemann, Damgaard,
Magnea and Nishigaki, the spectral density of a chRMT at nonzero temperature
and the Stephanov solution for chRMT at nonzero chemical potential.
