  It is generally accepted that statistics of energy levels in closed chaotic
quantum systems is adequately described by the theory of Random Hermitian
Matrices. Much less is known about properties of "resonances" - generic
features of open quantum systems pertinent for understanding of scattering
experiments.
  In the framework of the Heidelberg approach to quantum chaotic scattering
open systems are characterized by an effective non-Hermitian random matrix
Hamiltonian $\hat{H}_{ef}$. Complex eigenvalues of $\hat{H}_{ef}$ are S-matrix
poles (resonances). We show how to find the mean density of these poles
(Fyodorov and Sommers) and how to use the effective Hamiltonian to calculate
autocorrelations of the photodissociation cross section (Fyodorov and
Alhassid).
  In the second part of the paper we review recent results (Fyodorov,
Khoruzhenko and Sommers) on non-Hermitian matrices with independent entries in
the regime of weak Non-Hermiticity. This regime describes a crossover from
Hermitian matrices characterized by Wigner-Dyson statistics of real eigenvalues
to strongly non-Hermitian ones whose complex eigenvalues were studied by
Ginibre.
