Graphs are indispensable data structures used to model interconnected nodes and their relationships. However, devising filters for graph data is challenging. Previous approaches, including BernNet, utilized Bernstein polynomials to approximate spectral graph convolutions but suffered from lengthy training times compared to alternatives. To address this, we propose TaylorNet, a novel graph neural network built upon rigorous mathematical principles and theoretical foundations. TaylorNet provides an efficient approach for learning arbitrary graph spectral filters. Our methodology revolves around utilizing Kth order Taylor polynomials to estimate filters on the normalized Laplacian operator of graphs. By carefully setting the coefficients of these polynomials, we achieve effective filter design. Additionally, TaylorNet personalizes filter learning for specific graphs by leveraging observed graph data and signals to determine these coefficients. Extensive experimentation demonstrates TaylorNet’s ability to learn arbitrary spectral filters, surpassing current state-of-the-art methods, particularly for handling large graphs. Its exceptional performance extends to real-world graph modeling tasks, highlighting its versatility and applicability. For interested researchers, the complete implementation of TaylorNet is available at https://github.com/12chen20/TaylorNet.