Questions of stability for groups ask whether almost representations of a group have to be close to actual representations. Here, "almost" and "close" are measured by different norms. Such questions are related to approximation problems for groups. While for Hilbert-Schmidt-type norms, a lot of work had been done in recent years, stability with respect to the operator norm remained quite mysterious. In a recent joint work with Soren Eilers and Adam Sorensen we initiated a systematic study of the operator-norm stability. In this talk I will explain some methods for exploring stability that we developed, and show how they allow to obtain results on the stability/non-stability of various classes of discrete groups. Joint work with Eilers and Sorensen.