In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety "X" over a field F with a transitive action of a reductive group "G" (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If "X" has an F-rational point, then it is isomorphic to "G"/"P" for some parabolic subgroup "P" of "G". A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of "G". The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of "G", and they are precisely the coadjoint orbits of compact Lie groups.