  These lectures provide an informal introduction into the notions and tools
used to analyze statistical properties of eigenvalues of large random Hermitian
matrices. After developing the general machinery of orthogonal polynomial
method, we study in most detail Gaussian Unitary Ensemble (GUE) as a
paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics
of Hermite polynomials in various regimes and employ it in spectral analysis of
the GUE. In the last part of the course we discuss general relations between
orthogonal polynomials and characteristic polynomials of random matrices which
is an active area of current research.
