Variational integrators preserve geometric and topological structure when applied to Hamiltonian systems. Most of the research into variational integrators has focused upon their derivation by discretizing Hamilton's principle as a type I generating function of the symplectic map. In this talk we examine the derivation of variational integrators from a type II generating function. Even when the maps resulting from different generating functions are analytically equivalent there can be important numerical differences. We introduce a new class of variational integrators based on the Taylor method and an augmented shooting method. The role of automatic differentiation for an efficient implementation is discussed. Finally, a new framework for adaptive variational integrators is presented, which is dependent upon Hamiltonian variational integrators.