  We develop a machinery of Chen iterated integrals for higher Hochschild
complexes. These are complexes whose differentials are modeled on an arbitrary
simplicial set much in the same way the ordinary Hochschild differential is
modeled on the circle. We use these to give algebraic models for general
mapping spaces and define and study the surface product operation on the
homology of mapping spaces of surfaces of all genera into a manifold. This is
an analogue of the loop product in string topology. As an application, we show
this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type
theorems and use them to give explicit formulae for the surface product of odd
spheres and Lie groups.
