Homological stability has emerged over the past decades as an organizing principle in topology and beyond. Broadly speaking, many sequences of moduli spaces exhibit the striking phenomenon that their homology stabilizes as the underlying complexity grows. One of the most remarkable achievements of this paradigm is the proof of the Mumford conjecture by Madsen and Weiss, which identifies the stable rational cohomology of the moduli space of algebraic curves. This resolves a question of central importance in algebraic geometry and geometric group theory; all existing proofs use significant amounts of homotopy theory. In this lecture I will survey some ideas behind homological stability and the ubiquity of stability phenomena, and discuss consequences and applications, particularly in algebraic geometry and number theory.