The concepts of certain second-order structures on finite dimensional manifolds, Hessian structures and sprays are introduced in [6]. It is shown that each of these structures is uniquely determined by a symmetric connection. In this thesis we extend these results to Banach manifolds. We define Christoffel structures which is central to our exposition and yields all the results. Moreover we define nth order Hessian structures on manifolds and study them. In particular, when n=3, we make a detailed study and establish a one-to-one correspondence between third-order Hessian structures and certain Also, we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures. Finally, we extend all of these results to Frechet manifolds. 2000 MSC : Primary 35Q53. Secondary 37K30, 37K65. Keywords : Hessian structure, Christoffel structure, connection, spray.