A method for simulating fluid flows on curved surfaces is presented. Previous methods required flattening the curved surface to perform a standard 2d simulation. This flattening, however, introduces distortions to the flow. The proposed method avoids flattening and operates directly on the curved surface given as triangle mesh. Physical soundness of the simulation results and numerical consistency for different discretizations are shown for the proposed method. The method can be applied for simulating thin film and coating fluid flows, where the fluid is constrained to reside on the curved surface of an object or for simplifying full three-dimensional fluid flows to curved surfaces.