We present two models for Arnold diffusion in the three-body problem. 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. Equivalently, there exist trajectories that start near the Lyapunov orbit of the unperturbed problem for some given energy, and end up near the Lyapunov orbit of the unperturbed problem for some other given energy. 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 heteroclinic connections to this manifold. 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.