  This is a survey on cohomogeneity one manifolds with positive curvature. We
discuss the known examples of this type and their geometry and the functions
that describe the metric. We also describe the classification of cohomogeneity
one manifolds that can admit a metric with positive curvature due to
Grove-Wilking-Ziller. Two series of candidates arose in this classification for
which it is not yet know if they admit positive curvature. The connection of
these candidates to the Hitchin self dual Einstein orbifolds is discussed as
well, including the curvature properties of the Hitchin metrics.
