We study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any n \geq 2. We show that the automorphism group of S^{[n]} is either trivial or generated by a non-symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. This provides a generalization of results by Boissi\`ere--Cattaneo--Nieper-Wisskirchen--Sarti for n=2.
