Résumé : The aims of this talk are two-fold: a) to give a biased overview of some structural properties of (smooth) symplectic mapping class groups, together with some key proof ingredients, and b) to present some open questions, both about those groups and about their $C^{0}$ counterparts, whose study was recently initiated by Alexandre Jannaud.

Keywords : symplectic topology; mapping class groups; Dehn twists