We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete $p$-toral groups with an abelian subgroup of index $p$. In all of these cases (finite or infinite), we reduce the problem to one of listing all $\mathbb{F}_pG$-modules (for $G$ finite) satisfying certain conditions: a problem which was solved in the earlier paper by Craven, Oliver, and Semeraro using the classification of finite simple groups.