  We give an exhaustive description of all simply connected odd dimensional
cohomogeneity one manifolds that can possibly support an invariant metric with
positive sectional curvature. Among the known examples of odd dimensional
manifolds with positive curvature, apart from spheres, there are two infinite
families among the 7-dimensional Eschenburg spaces and 13-dimensional Bazaikin
spaces and one isolated 7-dimensional Berger space with this property. In
addition to these examples, it turns out that only one isolated 7-manifold, and
two infinite 7-dimensional families, potentially admit invariant cohomogeneity
one metrics of positive curvature.
