We will describe a  geometric method to prove instability in nearly integrable Hamiltonian systems of n-degrees of freedom. The approach is based on tracking the `outer dynamics’ along homoclinic orbits  to a normally hyperbolic invariant manifold (NHIM). Only little information is needed on the `inner dynamics' restricted to the NHIM, so this applies to rather general situations; for instance, the unperturbed Hamiltonian does not need to be convex.  The   conditions needed for this approach are transversality conditions and hence generic. Moreover, these conditions can be verified in concrete systems, such as  in celestial mechanics.