Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem  and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.

In this talk, I will survey some of the main results and and questions in this area, with a focus on cohomological techniques based on joint works with Glebsky, Lubotzky, Monod, and Fournier-Facio.