Title: Symmetries of manifolds

Abstract: Whenever one studies a mathematical object one ought also to study its symmetries. Manifolds are the central objects of study in topology and geometry, and their groups of symmetries come in many flavours (isometries, diffeomorphisms, homeomorphisms, …). I will discuss some classical and recent results about the spaces of all symmetries of certain simple manifolds, and report on an emerging conjectural picture.