In the first half of the talk I will talk about automorphism groups of varieties, focusing in a second moment on irreducible symplectic varieties. In the second part of the talk I will present a result obtained jointly with L. Giovenzana, A. Grossi and D. Veniani on the (non-)existence of symplectic automorphisms on irreducible symplectic varieties obtained as symplectic desingularisation of moduli spaces of sheaves on K3 surfaces.