  Starting with Arnold's pioneering work, the term "Arnold diffusion" has been
used to describe the slow diffusion taking place in the space of the actions in
Hamiltonian nonlinear dynamical systems with three or more degrees of freedom.
The present text is an elaborated transcript of the introductory course given
in the Milano I-CELMECH school on the topic of Arnold diffusion and its
relation to Nekhoroshev theory. The course introduces basic concepts related to
our current understanding of the mechanisms leading to Arnold diffusion.
Emphasis is placed upon the identification of those invariant objects in phase
space which drive chaotic diffusion, such as the stable and unstable manifolds
emanating from (partially) hyperbolic invariant objects. Besides a qualitative
understanding of the diffusion mechanisms, a precise quantification of the
speed of Arnold diffusion can be achieved by methods based on canonical
perturbation theory, i.e. by the construction of a suitable normal form at
optimal order. As an example of such methods, we discuss the
(quasi-)stationary-phase approximation for the selection of remainder terms
acting as driving terms for the diffusion. Finally, we discuss the efficiency
of such methods through numerical examples in which the optimal normal form is
determined by a computer-algebraic implementation of a normalization algorithm.
