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We present a general mechanism to establish the existence of diffusing orbitsin a large class of nearly integrable Hamiltonian systems. Our approach relieson successive applications of the `outer dynamics' along homoclinic orbits to anormally hyperbolic invariant manifold. The information on the outer dynamicsis encoded by a geometrically defined map, referred to as the `scattering map'.  We find pseudo-orbits of the scattering map that keep advancing in someprivileged direction. Then we use the recurrence property of the `innerdynamics', restricted to the normally hyperbolic invariant manifold, to returnto those pseudo-orbits. Finally, we apply topological methods to show theexistence of true orbits that follow the successive applications of the twodynamics.  This method differs, in several crucial aspects, from earlier works. Unlikethe well known `two-dynamics' approach, the method we present relies on theouter dynamics alone. There are virtually no assumptions on the inner dynamics,as its invariant objects (e.g., primary and secondary tori, lower dimensionalhyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are notused at all.  The method applies to unperturbed Hamiltonians of arbitrary degrees offreedom that are not necessarily convex. In addition, this mechanism is easy toverify (analytically or numerically) in concrete examples, as well as toestablish diffusion in generic systems.  We include several applications, such as bridging large gaps in a prioriunstable models in any dimension, and establishing diffusion in cases when theinner dynamics is a non-twist map.

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results