An algorithm is presented that finds the shortest path between two points on a polyhedral surface in O(n5) time, where n is the number of vertices on the surface, thereby establishing that the problem can be solved in polynomial time. The path is confined to the surface, and shortest is defined in terms of Euclidean distance. The algorithm is an extension of methods developed by Sharir and Schorr for finding the shortest path on a convex polyhedron. We relax the convexity assumption, only requiring that the surface be orientable.