  For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset
G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold
$N$ the correspondence space $\Cal CN$, which is the total space of a fiber
bundle over $N$ with fiber a generalized flag manifold, and construct a
canonical parabolic geometry of type $(G,Q)$ on $\Cal CN$.
  Conversely, for a parabolic geometry of type $(G,Q)$ on a smooth manifold
$M$, we construct a distribution corresponding to $P$, and find the exact
conditions for its integrability. If these conditions are satisfied, then we
define the twistor space $N$ as a local leaf space of the corresponding
foliation. We find equivalent conditions for the existence of a parabolic
geometry of type $(G,P)$ on the twistor space $N$ such that $M$ is locally
isomorphic to the correspondence space $\Cal CN$, thus obtaining a complete
local characterization of correspondence spaces.
  We show that all these constructions preserve the subclass of normal
parabolic geometries (which are determined by some underlying geometric
structure) and that in the regular normal case, all characterizations can be
expressed in terms of the harmonic curvature of the Cartan connection, which is
easier to handle. Several examples and applications are discussed.
