Let M be a smooth manifold of dimension n with smooth Riemannian metric g and u = (u₁,ĵ,u_N), n < N, be a smooth mapping of M into euclidean N-space. u is an isometric embedding if <du, du> = g. In terms of local coordinates this is a system of nonlinear partial differential equations of first order, which is overdetermined if N > 1/2 n (n+1). The local isometric embedding problem is to prove the existence, the uniqueness (rigidity) and the regularity of solutions. This problem gave birth to two theories of fundamental importance in nonlinear analysis. One is the theory of exterior differential system that was originated by E. Cartan in 1920's and further developed by Kaeler, Kuranishi, Chern, Griffiths, Bryant and others. This method solves the existence in analytic category and the rigidity problems. The other is the iteration scheme analogous to Newton's method for finding roots of a polynomial. This was initiated by J. Nash in 1950's and developed later to be the Nash-Moser implicit function theorem. These methods have far reaching applications to nonlinear and overdetermined systems beyond isometric embedding problem. In this talk we introduce after a brief survey of history a new approach to the problem, namely, complete prolongation of overdetermined systems and checking the integrability of the associated Pfaffian systems.