Intersection (co)homology is a beautiful invariant of singular spaces, e.g. singular complex algebraic varieties. Over the rationals it satisfies a raft of strong properties (e.g. Poincaré duality, hard Lefschetz, pure Hodge structure,…) which still seem like a miracle. In their paper introducing these groups, Goresky and MacPherson point out that (unlike the integral cohomology of smooth manifolds) it fails to satisfy Poincaré duality over the integers. Subsequent sheaf theoretic approaches have clarified the situation, and in fact one can prove that there is no reasonable theory over the integers which satisfies Poincaré duality. I will discuss intersection cohomology, given several simple examples that illustrate the theory. Then I will discuss work in progress with F. Klein and T. Fell where we introduce self-dual theory over certain p-adic rings. Interestingly, a self-dual theory exists if one is allowed to add enough roots of p. We don’t quite know what to make of this, but maybe someone in the audience does…