This chapter serves as an introduction to the book, providing a systematic review of the development of neural networks since the McCulloch-Pitts model was proposed in the 1940s, dividing it into three key phases. It highlights the core characteristics of neural networks, such as high fault tolerance, parallel processing capabilities, self-learning and self-adaptability, and powerful nonlinear approximation. The chapter extensively enumerates the broad applications of neural networks across various fields including image processing, signal analysis, pattern recognition, robotic control, healthcare, and economic forecasting, underscoring their utility as versatile tools. To establish the theoretical foundation for subsequent practical implementation, the chapter focuses on analyzing and comparing five fundamental training algorithms: Gradient Descent, Newton’s Method, the Conjugate Gradient Method, the Quasi-Newton Method, and the Levenberg-Marquardt algorithm. Using flowcharts and mathematical formulations, it explains their respective principles, advantages, disadvantages, and suitable application scenarios, offering clear guidance for readers to select the appropriate optimization method based on specific problems.