  In this paper we give a spinorial representation of submanifolds of any
dimension and codimension into Riemannian space forms in terms of the existence
of so called generalized Killing spinors. We then discuss several applications,
among them a new and concise proof of the fundamental theorem of submanifold
theory. We also recover results of T. Friedrich, B. Morel and the authors in
dimension 2 and 3.
