We study non-reversible Finsler metrics with constant flag curvature 1 on S2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1- parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S²with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.