This is an important contribution to nonlinear Fredholm theory. It is one in a series of papers by the authors motivated by needs in symplectic field theory (SFT) for generalizing methods to compactify solution sets to nonlinear Fredholm operators. The resulting geometric objects by such compactifications are called polyfolds and like manifolds, polyfolds have good properties in compactness and transversality issues. In the present paper, the authors establish a rigorous basis for transversality arguments in SFT by developing an integration theory for differential forms over polyfolds including a suitable Stokes' theorem. The paper is by nature rather technical but it is well written.