  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.
