  In this article, we first describe a normal form of real-analytic,
Levi-nondegenerate submanifolds of $C^N$ of codimension d $\ge$ 1 under the
action of formal biholomorphisms, that is, of perturbations of
Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal
normal form that ensures that the normalizing transformation to this normal
form is holomorphic. We show that our techniques can be adapted in the case d =
1 in order to obtain a new and direct proof of Chern-Moser normal form theorem.
