  In this paper, we study formal mappings between smooth generic submanifolds
in multidimensional complex space and establish results on finite
determination, convergence and local biholomorphic and algebraic equivalence.
Our finite determination result gives sufficient conditions to guarantee that a
formal map as above is uniquely determined by its jet at a point of a
preassigned order. For real-analytic generic submanifolds, we prove convergence
of formal mappings under appropriate assumptions and also give natural
geometric conditions to assure that if two germs of such submanifolds are
formally equivalent, then they are necessarily biholomorphically equivalent. If
the submanifolds are moreover real-algebraic, we address the question of
deciding when biholomorphic equivalence implies algebraic equivalence. In
particular, we prove that if two real-algebraic hypersurfaces in $\C^N$ are
biholomorphically equivalent, then they are in fact algebraically equivalent.
All the results are first proved in the more general context of ``reflection
ideals" associated to formal mappings between formal as well as real-analytic
and real-algebraic manifolds.
