Dan Petersen of the Institute for Advanced Study surveys homological stability, the phenomenon where the homology of sequences of moduli spaces settles into a fixed pattern as complexity grows. He traces the paradigm back to Madsen and Weiss's proof of the Mumford conjecture, which pins down the stable rational cohomology of mapping class groups using tools from homotopy theory, and explains why this result mattered to both algebraic geometry and geometric group theory. The talk moves from the general machinery behind stability arguments to concrete applications, particularly how these ideas feed back into algebraic geometry and number theory. Delivered as an IAS Members' Colloquium talk, it assumes some background in topology but frames the big picture clearly enough for a broader mathematical audience, emphasizing why so many different families of spaces exhibit this same stabilizing behavior.