Abstract: A finite group of symplectic automorphisms of a K3 surface is a subgroup of the Mathieu group M23 with certain numerical condition (Mukai, 1988). Recently there are systematic studies of automorphisms of hyper-Kahler manifolds by Mongardi, Hohn, Mason, etc. Cubic fourfolds are closely related to hyper-Kahler manifolds. In this talk I will sketch a complete classification of symplectic automorphism groups of smooth cubic fourfolds. This is a joint work with Radu Laza.