  We study spin structures on compact simply-connected homogeneous
pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G.
We classify flag manifolds F = G/H of a compact simple Lie group which are
spin. This yields also the classification of all flag manifolds carrying an
invariant metaplectic structure. Then we investigate spin structures on
principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently
simply-connected homogeneous complex manifolds M=G/L of a compact semisimple
Lie group G. We study the topology of M and we provide a sufficient and
necessary condition for the existence of an (invariant) spin structure, in
terms of the Koszul form of F. We also classify all C-spaces which are fibered
over an exceptional spin flag manifold and hence they are spin.
