Shape descriptors have recently gained popularity in shape matching, statistical shape mode-ling, etc. Their discriminative ability and efficiency play a decisive role in these tasks. In this paper, we first propose a novel handcrafted anisotropic spectral descriptor using Chebyshev polynomials, called the anisotropic Chebyshev descriptor (ACD); it can effec-tively capture shape features in multiple directions. The ACD inherits many good characteristics of spectral descriptors, such as being intrinsic, robust to changes in surface discretization, etc. Furthermore, due to the orthogonality of Chebyshev polynomials, the ACD is compact and can disambiguate intrinsic symmetry since several directions are considered. To improve the ACD’s discrimination ability, we construct a Chebyshev spectral manifold convolutional neural network (CSMCNN) that optimizes the ACD and produces a learned ACD. Our experimental results show that the ACD outperforms existing state-of-the-art handcrafted descriptors. The combination of the ACD and the CSMCNN is better than other state-of-the-art learned descriptors in terms of discrimination, efficiency, and robustness to changes in shape resolution and discretization.