We present an algorithm for finding a meaningful vertexto- vertex correspondence between two 3D shapes given as triangle meshes. Our algorithm operates on embeddings of the two shapes in the spectral domain so as to normalize them with respect to uniform scaling and rigid-body transformation. Invariance to shape bending is achieved by relying on geodesic point proximities on a mesh to capture its shape. To deal with stretching, we propose to use non-rigid alignment via thin-plate splines in the spectral domain. This is combined with a refinement step based on the geodesic proximities to improve dense correspondence. We show empirically that our algorithm outperforms previous spectral methods, as well as schemes that compute correspondence in the spatial domain via non-rigid iterative closest points or the use of local shape descriptors, e.g., 3D shape context.