The purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex CR structures over small balls along the line developed by D. C. Spencer, C. B. Morrey, J. J. Kohn and L. Nirenberg ([5], [6], [7], [8]) for strongly pseudoconvex complex manifolds. We consider a strongly pseudoconvex CR structure on a manifold of real dimension 2n-1. In Part I we establish the a priori estimate for the Neumann boundary value problem on the complex associated with the structure, in the case the structure is induced by an embedding in C and restricted to a small ball of special type, provided I c q c n 3 where q is the degree of differential forms we consider. In Part II we derive the regularity theorem of solutions of the Neumann boundary value problem based on the a priori estimate of Part I. As an application we prove in Part Il that, when n ? 5, the structure is realized on a neighborhood of a reference point by an embedding in Cn. We note here that L. Nirenberg gave an example of a strongly pseudoconvex CR structure which cannot be induced by such an embedding when n =2, [9]. L. Boutet de Monvel proved that, in the case of such structures over a compact manifold, we can find such an embedding when n -3, [1]. There are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and CR structures. Firstly the principal parts of the associated Laplacians in CR structures are no longer elliptic. Secondly, we have characteristic points on the boundary. As a result, when we follow over the predecessor's calculation in our case, we end up with terms which may not be absorbed in the main non-negative terms. Fortunately these leftover terms depend mostly on the Levi-form of the CR structure and the hessian form (with respect to the CR structure) of the equation of the boundary of our