Un th\'eor\`eme de Kerckhoff, Masur et Smillie : Unique ergodicit\'e sur les surfaces plates

This article is based on the lectures given at the ``Ecole thematique de theorie ergodique'', at the C.I.R.M. in Marseille, in April 2006. We give a complete proof of a theorem of Kerckhoff, Masur and Smillie on the unique ergodicity of the directional flow on a translation surface in almost every direction. The proof follows the one presented in the survey of Masur and Tabachnikov.