  We survey a (nonlinear) Fredholm theory for a new class of ambient spaces
called polyfolds, and develop the analytical foundations for some of the
applications of the theory. The basic feature of these new spaces, which can be
finite and infinite dimensional, is that in general they may have locally
varying dimensions. These new spaces are needed for a functional analytic
treatment of nonlinear problems involving analytic limiting behavior like
bubbling-off. The theory is applicable to Gromov-Witten and Floer Theory as
well as Symplectic Field Theory.
