The aim of this talk is to show that C*-algebras are useful for studying stability of groups. In particular we will discuss some obstructions for Hilbert-Schmidt stability of groups, obtain a complete characterization of Hilbert-Schmidt stability for amenable groups in terms of characters, and discuss some examples of amenable and non-amenable Hilbert-Schmidt stable groups. If time permits we also will consider certain modifications of Hilbert-Schmidt stability when matrices are replaced by elements of von Neumann factors, as well as a relation between Hilbert-Schmidt stability and operator norm-stability. All necessary information on C*-algebras will be given during the talk. The talk is mostly based on joint works with Don Hadwin.