The Borel conjecture predicts that closed, aspherical manifolds (i.e., those with contractible universal cover) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth version of this conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds.