A cohomogeneity one manifold is a manifold with the action of a compact Lie
group, whose quotient is one dimensional. Such manifolds are of interest in
Riemannian geometry, in the context of nonnegative sectional curvature, as well
as in other areas of geometry and in physics. In this paper we classify compact
simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also
show that all such manifolds admit metrics of nonnegative sectional curvature,
with the possible exception of two families of manifolds
