  The twistor construction for Riemannian manifolds is extended to the case of
manifolds endowed with generalized metrics (in the sense of generalized
geometry \`a la Hitchin). The generalized twistor space associated to such a
manifold is defined as the bundle of generalized complex structures on the
tangent spaces of the manifold compatible with the given generalized metric.
This space admits natural generalized almost complex structures whose
integrability conditions are found in the paper. An interesting feature of the
generalized twistor spaces discussed in it is the existence of intrinsic
isomorphisms.
