We present a method for the adaptive reconstruction of a surface directly from an unorganized point cloud. The algorithm is based on an incrementally expanding Neural Network and the statistical analysis of its Learning process. In particular, we make use of the simple observation that during the Learning process the normal of a vertex near a sharp edge or a high curvature area of the target space, statistically, will vary more than the normal of a vertex near a flat area. We use the information obtained from the study of these normal variations to steer the Learning process in an adaptive meshing application, producing meshes with more triangles near the high curvature areas. The same information is used in a feature detection application.