Let M be an oriented manifold; (higher) string topology studies the algebraic structure of the homology of the free loop manifold as well as free sphere manifolds Map(S^n,M). There is a standard isomorphism between the homology of the free loop space of a simply connected space and the Hochschild cohomology of its singular cochains algebra. We will explain how to generalize the latter isomorphism to higher sphere manifolds at the cochain level using a higher generalization of Hochschild (co)homology and how this equivalence allows to endow the chains on the free sphere manifold Map(S^n,M) with an E_{n+1}-algebra structure.