  Iterated Segre mappings of real analytic generic submanifolds in complex
space have been an essential tool in the study of holomorphic, formal, and CR
mappings between such manifolds. In this paper we present a theory of iterated
Segre mappings for smooth or formal submanifolds. In particular, we
characterize finite type in terms of the rank of the iterated Segre mappings.
The approach given here is new, simplifies proofs, and extends results
previously known for the real analytic case.
