  We prove here new results about transversality and related geometric
properties of a holomorphic, formal, or CR mapping, sending one generic
submanifold of $\bC^N$ 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.
