  Assuming the validity of random matrices for describing the statistics of a
closed chaotic quantum system, we study analytically some statistical
properties of the S-matrix characterizing scattering in its open counterpart.
In the first part of the paper we attempt to expose systematically ideas
underlying the so-called stochastic (Heidelberg) approach to chaotic quantum
scattering. Then we concentrate on systems with broken time-reversal invariance
coupled to continua via M open channels. By using the supersymmetry method we
derive:
  (i) an explicit expression for the density of S-matrix poles (resonances) in
the complex energy plane
  (ii) an explicit expression for the parametric correlation function of
densities of eigenphases of the S-matrix.
  We use it to find the distribution of derivatives of these eigenphases with
respect to the energy ("partial delay times" ) as well as with respect to an
arbitrary external parameter.
