We study the stationary, ideal flow on a free fluid surface with a prescribed shape. It is demonstrated that the flow is governed by a self-contained set of equations for the surface flow field without any reference to the bulk flow. To write down these equations for arbitrary surfaces, we apply a covariant formulation using Riemannian geometry and we show how to include surface tension and velocity-dependent forces such as the Coriolis force. We write down explicitly the equations for cases where the surface elevation can be written as function of either Cartesian or polar coordinates in the plane, and we obtain solutions for the important case of rotational symmetry and the perturbed flow when this symmetry is slightly broken. To understand the general character and solubility of the equations, we introduce the associated dynamical system describing the motion along the streamlines. The existence of orbits with transversal intersections, as well as quasi-periodic and chaotic solutions, show that not all boundary value problems are well-posed. In the particular case of unforced motion the streamlines are geodesic curves and in this case the existence of a non-trivial surface velocity field requires that the surface can be foliated by a family of non-intersecting geodesic curves.