  Random Matrix Theory has been a unifying approach in physics and
mathematics.In these lectures we discuss applications of Random Matrix Theory
to QCD and emphasize underlying integrable structures. In the first lecture we
give an overview of QCD, its low-energy limit and the microscopic limit of the
Dirac spectrum which, as we will see in the second lecture, can be described by
chiral Random Matrix Theory. The main topic of the third lecture is the recent
developments on the relation between the QCD partition function and integrable
hierarchies (in our case the Toda lattice hierarchy). This is an efficient way
to obtain the QCD Dirac spectrum from the low energy limit of the QCD partition
function. Finally, we will discuss the QCD Dirac spectrum at nonzero chemical
potential. We will show that the microscopic spectral density is given by the
replica limit of the Toda lattice equation. Recent results by Osborn on the
Dirac spectrum of full QCD will be discussed.
