  This volume contains a mildly expanded version of lectures and talks at
seminars and conferences, as well as review papers on subjects listed in the
title of the volume. A great deal of these texts have already been published or
sent to press. However, the only way to provide a good exposition of the field
we are interested in is to collect all of those papers together.
  Bounded symmetric domains form a favorite subject of research in function
theory, non-commutative harmonic analysis, and representation theory of real
reductive Lie groups. The authors introduce the notions of q-analogues of
bounded symmetric domains and q-Harish-Chandra modules. For that, they follow
the traditions of quantum group theory in replacing Lie groups with the quantum
universal enveloping Drinfeld-Jimbo algebras and representations of groups with
associated Harish-Chandra modules.
  The purpose of this volume is to convince the reader in exceptional
attraction of the deduced class of quantum homogeneous spaces and the related
class of modules over quantum universal enveloping algebras.
