A robust statistics approach to curvature estimation on discretely sampled surfaces, namely polygon meshes and point clouds, is presented. The method exhibits accuracy, stability and consistency even for noisy, non-uniformly sampled surfaces with irregular configurations. Within an M-estimation framework, the algorithm is able to reject noise and structured outliers by sampling normal variations in an adaptively reweighted neighborhood around each point. The algorithm can be used to reliably derive higher order differential attributes and even correct noisy surface normals while preserving the fine features of the normal and curvature field. The approach is compared with state-of-the-art curvature estimation methods and shown to improve accuracy by up to an order of magnitude across ground truth test surfaces under varying tessellation densities and types as well as increasing degrees of noise. Finally, the benefits of a robust statistical estimation of curvature are illustrated by applying it to the popular applications of mesh segmentation and suggestive contour rendering. <p></p> <b><a href="https://graphics.stanford.edu/~kalo/papers/curvature/index.html">Project web page</a></b>