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We present a new multiscale surface representation for 3D shape matching that is based on scale-space theory. The representation, Curvature Scale-Space 3D (CS3), is well-suited for measuring dissimilarity between (partial) surfaces having unknown position, orientation, and scale. The CS3 representation is obtained by evolving the surface curvatures according to the heat equation. This evolution process yields a stack of increasingly smoothed surface curvatures that is useful for keypoint extraction and descriptor computations. We augment this information with an associated scale parameter at each stack level to define our multiscale CS3 surface representation. The scale parameter is necessary for automatic scale selection, which has proven to be successful in 2D scale-invariant shape matching applications. We show that our keypoint and descriptor computation approach outperforms many of the leading methods. The main advantages of our representation are its computational efficiency, lower memory requirements, and ease of implementation.