Let M be an even-dimensional Riemannian manifold, the twistor
space Z(M) is the parametrizing space for compatible almost complex
structures on M. We construct the twistor space of the normal bundle of
a foliation. It is demonstrated that the classical constructions of the
twistor theory lead to foliated objects and permit formulations and
proofs of foliated versions of some well-known results on holomorphic
mappings. Since any orbifold can be understood as the leaf space of a
suitably defined Riemannian foliation, we obtain orbifold versions of
the classical results as a simple consequence of the results on foliated
mappings.