Transversality of CR mappings

We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of C^N into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism.