The appropriate context for algebraic-geometric realizations of holomorphic representations of a complex semisimple group is that of a compact homogeneous projective flag manifold Z = G/Q. One topic of more current interest is the study of the possibility of realizing (infinite-dimensional) unitary representations of a real form G0 of G on function- and/or cohomology-spaces of open G_0-orbits D in Z (flag domains) and their cycle spaces. After an introduction for nonspecialists, we will indicate a proof by Schubert incidence geometry of the Kobayashi hyperbolicity of the relevant cycle spaces. This will then be applied to give an exact description of the group Aut_O(D) of holomorphic automorphisms.