Many representative graph neural networks, e.g., GPR-GNN and ChebNet,\napproximate graph convolutions with graph spectral filters. However, existing\nwork either applies predefined filter weights or learns them without necessary\nconstraints, which may lead to oversimplified or ill-posed filters. To overcome\nthese issues, we propose BernNet, a novel graph neural network with theoretical\nsupport that provides a simple but effective scheme for designing and learning\narbitrary graph spectral filters. In particular, for any filter over the\nnormalized Laplacian spectrum of a graph, our BernNet estimates it by an\norder-$K$ Bernstein polynomial approximation and designs its spectral property\nby setting the coefficients of the Bernstein basis. Moreover, we can learn the\ncoefficients (and the corresponding filter weights) based on observed graphs\nand their associated signals and thus achieve the BernNet specialized for the\ndata. Our experiments demonstrate that BernNet can learn arbitrary spectral\nfilters, including complicated band-rejection and comb filters, and it achieves\nsuperior performance in real-world graph modeling tasks. Code is available at\nhttps://github.com/ivam-he/BernNet.\n