Appears in collection : Foliations and Diffeomorphism Groups / Feuilletages et Groupes de Difféomorphisme

Unlike other similar groups, these groups remained very poorly understood for a long time. I'll review some recent progress in two directions: (non)-simplicity and the existence of quasimorphisms. I'll also give some ideas on the proofs which are based on tools from symplectic topology. This is based on joint works with Dan Cristofaro-Gardiner, Cheuk-Yu Mak, Sobhan Seyfaddini, and Ivan Smith.