ABSTRACT
After recalling few classical arithmetic results on continued fraction, a dynamical view is taken by introducing the Gauss map together with the famous invariant Gauss measure. The ergodic properties are thoroughly studied. As an application of the Pointwise Ergodic Theorem we provide a proof of L´evy's Theorem describing the asymptotic properties of these expansions. A planar version of the natural extension of the Gauss map is given and used to prove the Doeblin-Lenstra conjecture on the distribution of the approximation coefficients associated with continued fractions. The chapter ends with a discussion of other continued fraction expansions and a recently introduced random transformation generating all possible continued fraction expansions of real numbers.
