Most known smoothable simply connected 4–manifolds admit infinitely many different smooth structures (distinguished, for example, by Seiberg–Witten invariants). There are some 4–manifolds, though, for which the existence of such ‘exotic’ structures is still open, the most notable examples being the 4–dimensional sphere S4 and the complex projective plane CP2. In a recent project with Z. Szab´o and J. Park we found constructions of exotic smooth structures on the five- and six-fold blow-up of CP2. In the lecture we describe the construction of these 4–manifolds and indicate the necessary input from Seiberg–Witten theory for proving their exoticness.