In the first part of my talk I will discuss the problems of reconstructing a countable discrete group from its von Neumann algebra (W*-superrigidity) and its reduced C*-algebra (C*-superrigidity) and I will survey several recent results in this direction. In the second part, using and interplay between von Neumann algebraic and C*-algebraic methods, I will introduce a new class of C*-superrigid groups which appear as wreath products with non-amenable core. As an application we obtain complete calculations of the symmetry groups of various group C*-algebras---a problem barely touched in the literature. This is based on a recent joint work with Alec Diaz-Arias.