This comprehensive survey of Kolmogorov-Arnold Networks (KAN) provides a thorough understanding of its theoretical foundation, architectural design, application scenarios, and current research progress. KAN, based on the theorems of Kolmogorov and Arnold, represents continuous multivariable functions as superpositions of univariate functions, enhancing model flexibility and interpretability. Key features include learnable univariate functions for weights and dynamic activation functions, which allow KAN to handle complex data patterns and nonlinear relationships more effectively than traditional neural networks. The survey highlights KAN's advantages in various applications: 1. **Function Approximation and Multivariable Function Decomposition**: KAN excels in capturing complex nonlinear relationships and high-dimensional data, outperforming traditional methods in tasks like hyperspectral image classification and time series analysis. 2. **High-Dimensional Data Handling**: KAN's flexible architecture and dynamic activation functions make it particularly effective in processing large-scale and high-dimensional datasets. 3. **Enhanced Accuracy and Interpretability**: KAN's learnable activation functions improve model interpretability, making it suitable for critical applications such as medical diagnosis and financial forecasting. 4. **Training and Optimization**: Techniques like regularization, batch normalization, and advanced optimization algorithms (e.g., Adam and stochastic gradient descent) enhance KAN's training efficiency and generalization ability. 5. **Applications in Symbolic Regression, Time Series Prediction, Graph-Structured Data Processing, Hyperspectral Image Classification, Quantum Architecture Search, and Medical Image Segmentation**: KAN demonstrates superior performance in these areas, showcasing its broad applicability and potential for innovative solutions in various fields. Overall, KAN offers a promising approach to handling complex computational problems and is poised to revolutionize the field of machine learning and data science.This comprehensive survey of Kolmogorov-Arnold Networks (KAN) provides a thorough understanding of its theoretical foundation, architectural design, application scenarios, and current research progress. KAN, based on the theorems of Kolmogorov and Arnold, represents continuous multivariable functions as superpositions of univariate functions, enhancing model flexibility and interpretability. Key features include learnable univariate functions for weights and dynamic activation functions, which allow KAN to handle complex data patterns and nonlinear relationships more effectively than traditional neural networks. The survey highlights KAN's advantages in various applications: 1. **Function Approximation and Multivariable Function Decomposition**: KAN excels in capturing complex nonlinear relationships and high-dimensional data, outperforming traditional methods in tasks like hyperspectral image classification and time series analysis. 2. **High-Dimensional Data Handling**: KAN's flexible architecture and dynamic activation functions make it particularly effective in processing large-scale and high-dimensional datasets. 3. **Enhanced Accuracy and Interpretability**: KAN's learnable activation functions improve model interpretability, making it suitable for critical applications such as medical diagnosis and financial forecasting. 4. **Training and Optimization**: Techniques like regularization, batch normalization, and advanced optimization algorithms (e.g., Adam and stochastic gradient descent) enhance KAN's training efficiency and generalization ability. 5. **Applications in Symbolic Regression, Time Series Prediction, Graph-Structured Data Processing, Hyperspectral Image Classification, Quantum Architecture Search, and Medical Image Segmentation**: KAN demonstrates superior performance in these areas, showcasing its broad applicability and potential for innovative solutions in various fields. Overall, KAN offers a promising approach to handling complex computational problems and is poised to revolutionize the field of machine learning and data science.