We consider the problem of reconstructing a high-order surface from a given surface mesh. This problem is important for many meshing operations, such as generating highorder finite elements, mesh refinement, mesh smoothing and mesh adaptation. We introduce two methods, called Weighted Averaging of Local Fittings (WALF) and Continuous Moving Frames (CMF). These methods are both based on weighted least squares polynomial fittings and guarantee C 0 continuity. Unlike existing methods for

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... ting surfaces, our methods are applicable to surface meshes composed of triangles and/or quadrilaterals, can achieve third and even higher order accuracy, and have integrated treatments for sharp features. We present the theoretical framework of our methods, experimental comparisons against other methods, and its applications in a number of meshing operations.