A group G is called C*-superrigid if there is no other group with an isomorphic reduced C*-algebra. In 2010, Ioana, Popa and Vaes asked whether every discrete countable torsion-free group is C*-superrigid. In this talk, I will give an overview of the known examples of C*-superrigid groups and add certain classes of virtually abelian groups to the list. Finally, I will answer the question above by presenting examples of torsion-free groups which are not C*-superrigid. The proof uses recent results from classification theory of C*-algebras. This talk is based on several projects joint with subsets of Hua, Knudby, Raum, Slutsky, Thiel and White.
