Defining sharp features in a given 3D model facilitates a better understanding of the surface and aids visualizations, reverse engineering, filtering, simplification, non-photo realism, reconstruction and other geometric processing applications. We present a robust method that identifies sharp features in a point cloud by returning a set of smooth curves aligned along the edges. Our feature extraction is a multi-step refinement method that leverages the concept of Robust Moving Least Squares to locally fit surfaces to potential features. Using Newton's method, we project points to the intersections of multiple surfaces then grow polylines through the projected cloud. After resolving gaps, connecting corners, and relaxing the results, the algorithm returns a set of complete and smooth curves that define the features. We demonstrate the benefits of our method with two applications: surface meshing and point-based geometry compression.