This survey provides a comprehensive overview of Kolmogorov-Arnold Networks (KAN), covering their theoretical foundation, architectural design, application scenarios, and current research progress. KAN, based on the Kolmogorov-Arnold theorem, represents continuous multivariable functions as a superposition of univariate functions, offering flexibility and interpretability. Unlike traditional neural networks, KAN uses learnable univariate functions instead of fixed weights, enabling it to handle complex data patterns and nonlinear relationships effectively. While challenges remain, KAN shows great potential in various fields, including data fitting, partial differential equation solving, hyperspectral image classification, and time series analysis. The Kolmogorov-Arnold theorem, proposed by Andrey Kolmogorov and later extended by Vladimir Arnold, provides a mathematical foundation for KAN. It states that any continuous multivariable function can be represented as a finite superposition of univariate functions. This theorem has been applied in numerical analysis and partial differential equation solving, influencing the development of KAN. Recent advancements in KAN include the use of learnable univariate functions, dynamic activation functions, and parameter optimization techniques, enhancing its performance in complex tasks. KAN's architecture is unique, with each weight parameter represented by a learnable univariate function. This allows KAN to adapt to different data patterns and improve model flexibility and accuracy. KAN's dynamic activation functions enable it to adjust to various complex data patterns and nonlinear relationships. The architecture of KAN can be customized for specific tasks, offering flexibility in design. KAN has shown significant advantages in handling high-dimensional data, such as in time series analysis and graph-structured data processing. It has also demonstrated superior performance in hyperspectral image classification and quantum architecture search. KAN's ability to capture complex nonlinear relationships and its flexibility in parameter optimization make it a promising approach in various applications. The training process of KAN involves backpropagation and optimization techniques such as gradient descent, with challenges in ensuring stability and efficiency. Researchers have explored various strategies, including regularization, batch normalization, and advanced optimization algorithms, to enhance KAN's performance. These techniques help improve training efficiency and generalization ability, making KAN more effective in handling large-scale datasets. In summary, KAN offers a unique approach to neural network design, leveraging the Kolmogorov-Arnold theorem to represent complex functions and handle high-dimensional data. Its flexibility, interpretability, and performance in various applications make it a valuable tool in machine learning and data science. Future research directions include further exploring KAN's potential in different domains and improving its efficiency and scalability.This survey provides a comprehensive overview of Kolmogorov-Arnold Networks (KAN), covering their theoretical foundation, architectural design, application scenarios, and current research progress. KAN, based on the Kolmogorov-Arnold theorem, represents continuous multivariable functions as a superposition of univariate functions, offering flexibility and interpretability. Unlike traditional neural networks, KAN uses learnable univariate functions instead of fixed weights, enabling it to handle complex data patterns and nonlinear relationships effectively. While challenges remain, KAN shows great potential in various fields, including data fitting, partial differential equation solving, hyperspectral image classification, and time series analysis. The Kolmogorov-Arnold theorem, proposed by Andrey Kolmogorov and later extended by Vladimir Arnold, provides a mathematical foundation for KAN. It states that any continuous multivariable function can be represented as a finite superposition of univariate functions. This theorem has been applied in numerical analysis and partial differential equation solving, influencing the development of KAN. Recent advancements in KAN include the use of learnable univariate functions, dynamic activation functions, and parameter optimization techniques, enhancing its performance in complex tasks. KAN's architecture is unique, with each weight parameter represented by a learnable univariate function. This allows KAN to adapt to different data patterns and improve model flexibility and accuracy. KAN's dynamic activation functions enable it to adjust to various complex data patterns and nonlinear relationships. The architecture of KAN can be customized for specific tasks, offering flexibility in design. KAN has shown significant advantages in handling high-dimensional data, such as in time series analysis and graph-structured data processing. It has also demonstrated superior performance in hyperspectral image classification and quantum architecture search. KAN's ability to capture complex nonlinear relationships and its flexibility in parameter optimization make it a promising approach in various applications. The training process of KAN involves backpropagation and optimization techniques such as gradient descent, with challenges in ensuring stability and efficiency. Researchers have explored various strategies, including regularization, batch normalization, and advanced optimization algorithms, to enhance KAN's performance. These techniques help improve training efficiency and generalization ability, making KAN more effective in handling large-scale datasets. In summary, KAN offers a unique approach to neural network design, leveraging the Kolmogorov-Arnold theorem to represent complex functions and handle high-dimensional data. Its flexibility, interpretability, and performance in various applications make it a valuable tool in machine learning and data science. Future research directions include further exploring KAN's potential in different domains and improving its efficiency and scalability.