Abstract. 
We will describe the Arnold diffusion problem, asserting that generic,
nearly integrable Hamiltonian systems possess trajectories that travel
"wildly" and "arbitrarily far".

More precisely, one starts with a  Liouville integrable Hamiltonian
system; locally, this can be described via  action-angle coordinates.
Each trajectory  of the system lying within an action-angle domain
preserves the action coordinate indefinitely. Then one acts on the
integrable system with a small  perturbation of a generic type. The
main question is whether  there exist trajectories whose action
coordinate changes by some positive constant independent of the size
of the perturbation (diffusing orbits), as well as trajectories whose
action coordinate visits a prescribed collection of open sets in the
action space (symbolic dynamics), for all sufficiently small
perturbations.

We will describe a geometric/topological method, developed with R. de
la Llave and T. Seara,  that can be applied to prove the existence of
diffusing orbits and of symbolic dynamics for large classes of nearly
integrable Hamiltonian systems.