It is well-known that the derived category of a K3 surface is governed by its Mukai lattice. In this talk, I will discuss analogous phenomena for hyper-Kähler varieties of K3^[n]-type. In particular, I will show that a natural twisted derived category of a 2n-dimensional moduli space of stable objects on a K3 surface is derived equivalent to the Hilbert scheme of n points on the surface.