In this paper we address the problem of matching two 3D shapes by representing them using the eigenvalues and eigenvectors of the discrete diffusion operator. This provides representation framework useful for scale-space local shape descriptors and shape comparisons. We formally introduce diffusion embedding and we propose unit hypersphere normalizations of this embedding. We also propose a method to find compatible time scale while matching two different shapes with varying size and sampling. We propose a practical algorithm that seeks the largest set of mutually consistent point-to-point matches between two shapes based on isometric consistency between the two embedded shapes. We illustrate our method with several examples of matching shapes at various scales. © 2010 IEEE.