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.