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In this paper we introduce a new mesh filtering method: a mesh median filter. This is an application of the median filter to smoothen 3-D noisy shapes given by triangle meshes. An algorithm of the mesh median filter is realized by applying the median filter to face normals on triangle meshes and updating mesh vertex positions to make them fit to the filtered normals. As an advanced modification of the mesh median filter we further introduce a weighted mesh median filter. The weighted mesh median filter has a reinforced feature preservation effect. The weighted mesh median filter with positive weighting has the smoothing effect, and the one with negative weighting has the enhancing effect. The two kinds of mesh median filters are compared with two conventional mesh filtering methods: the Laplacian smoothing flow and the mean curvature flow. Experimental results demonstrate that the mesh median filter does not induce oversmoothing.