The broad context for this talk is the problem of classifying topological 4-manifolds, and the role of the fundamental group and of smooth structures in that problem. We will begin by recalling Mike Freedman's classification in the simply connected case, which says in particular that every non-singular integral symmetric bilinear form arises as the intersection form of *at most two* simply connected 4-manifolds. Most of these manifolds are not smoothable, by the work of Simon Donaldson. We will then describe, in contrast, how most of these forms arise as the intersection form of *infinitely many* non-smoothable 4-manifolds with infinite cyclic fundamental group. This is recent joint work with Friedl, Hambleton and Teichner.