## ABSTRACT

This chapter discusses the main ideas and techniques used in Arnold's proof of the KAM theorem. Some of them are traditional, like perturbation theory, some were relatively new at Arnold's time, like super-convergence, sets without interior, or a Diophantine condition used to overcome problems with small divisors, and some were new, like the cutoff of the Fourier series. In perturbation theory, unknown functions are expanded in power series with respect to a small parameter ε, and the equations of the problem are transformed into equations for the individual orders in ε. Motion on a phase-space torus for integrable systems was discussed in the chapter.