Abstract. 
We present two models for Arnold diffusion in celestial mechanics. The
first model is the spatial circular restricted three-body problem. We
show that, on some fixed energy level, there exist trajectories near
one of the libration points whose out-of-plane amplitude of motion
changes from nearly zero to nearly the maximum value for that energy
level. The second model is the planar elliptic restricted three-body
problem, with the eccentricity of the binary regarded as a
perturbation parameter. We show that there exist trajectories whose
energy changes between two given levels, for all sufficiently small
eccentricities. These are trajectories that start near the Lyapunov
orbit of the unperturbed problem at some energy level, and end up near
the Lyapunov orbit at some other energy levels. In both models we use
the existence of a normally hyperbolic invariant manifold, the "inner
dynamics" given by the restriction of the flow to this manifold, and
the "outer dynamics" given by the homoclinic connections to this
manifold (described via the scattering map). A key ingredient is a
topological shadowing lemma, which allows one to find true orbits near
pseudo-orbits given by alternately applying the inner dynamics and the
outer dynamics. This approach can be applied in both analytical
arguments and rigorous numerical experiments. This is based on joint
works with M. Capinski, A. Delshams, R. de la Llave, and P. Roldan.