Consideration is given to periodic traveling waves on the surface of a perfect fluid without the assumption that the flow is irrotational. If the variation in the direction transverse to the propagation of the traveling wave is negligible, the flow can be described by the two-dimensional Euler equations with appropriate boundary conditions. In this situation, two uniqueness results are presented. First, it is shown that in water of finite depth, the surface profile of a periodic traveling wave uniquely determines the corresponding flow in the body of the fluid. This holds for rotational flow as long as the vorticity function satisfies appropriate conditions which are also shown to be sharp. Second, attention is given to the case when the pressure is constant along streamlines. In this case, it is shown that the flow must be given by a special explicit solution which was found by Gerstner.