We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description
yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional
space and two infinite families not yet known to carry metrics of positive curvature. The infinite families also carry a 3-Sasakian
metric of cohomogeneity one, which is associated to a family of selfdual Einstein orbifold metrics on the 4-sphere constructed by
Hitchin.