  In contrast to the homogeneous case, we show that there are compact
cohomogeneity one manifolds, that do not support invariant metrics of
non-negative sectional curvature. In fact we exhibit infinite families of such
manifolds including the exotic Kervaire spheres. Such examples exist for any
codimension of the singular orbits except for the case where both are equal to
two, where existence of non-negatively curved metrics is known.
