Families of groups such as symmetric groups or general linear groups, and families of spaces such as con guration spaces or certain moduli spaces of manifolds, display a stability phenomenon in their homology. I'll describe this phenomenon and give an idea about how such theorems are proved. Reading List: First three sections of O. Randall-Willliams and N. Wahl, \Homological stability for autormorphism groups", Advances in Mathematics, Vol. 318 (2017), pp. 534 - 626.