  One of the most important questions in quantum information theory is the
so-called separability problem. It involves characterizing the set of separable
(or, equivalently entangled) states among mixed states of a multipartite
quantum system. In this thesis we study the generalization of this problem to
types of quantum correlations that are defined in a manner analogous to
entanglement. We start with the subset of set of pure states of a given quantum
system and call states belonging to the convex hull of this subset
"non-correlated" states. Consequently, the states laying outside the convex
hull are referred to as "correlated".
  In this work we focus on cases when there exist a symmetry group that
preserves the class of "non-correlated" pure states. The presence of symmetries
allows to obtain a unified treatment of many types of seemingly unrelated types
of correlations. We apply our general results to particular types of
correlations: (i) entanglement of distinguishable particles, (ii) particle
entanglement of bosons, (iii) "entanglement" of fermions, (iv)
non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty
entanglement, and finally (vi) refined notions of bipartite entanglement based
on the concept of the Schmidt number.
  We investigate the natural problems and questions concerning the correlations
defined above: (I) We provide explicit polynomial characterization of various
types of correlations for pure states. (II) We examine cases in which it is
possible to give a complete analytical characterization of correlated mixed
states. (III) We derive a variety of polynomial criteria for detection of
correlations in mixed states.(IV) We use the above criteria and the technique
of measure concentration to study typical properties of correlations on sets of
isospectral density matrices.
