In this paper, we propose a new and powerful shape denoising technique for processing surfaces approximated by triangle meshes and soups. Our approach is inspired by recent non-local image denoising schemes and naturally extends bilateral mesh smoothing methods. The main idea behind the approach is very simple. A new position of vertex P of a noisy mesh is obtained as a weighted mean of mesh vertices Q with nonlinear weights reflecting a similarity between local neighborhoods of P and Q. We demonstrate that our technique outperforms recent state-of-the-art smoothing methods. We also suggest a new scheme for comparing different mesh/soup denoising methods.