  This is an expanded version of a series of lectures delivered at the 25th
Winter School ``Geometry and Physics'' in Srni.
  After a short introduction to Cartan geometries and parabolic geometries, we
give a detailed description of the equivalence between parabolic geometries and
underlying geometric structures.
  The second part of the paper is devoted to constructions which relate
parabolic geometries of different type. First we discuss the construction of
correspondence spaces and twistor spaces, which is related to nested parabolic
subgroups in the same semisimple Lie group. An example related to twistor
theory for Grassmannian structures and the geometry of second order ODE's is
discussed in detail.
  In the last part, we discuss analogs of the Fefferman construction, which
relate geometries corresponding different semisimple Lie groups.
