This research paper provides a comprehensive review of methodologies for path planning and optimization of mobile robots. With the rapid development of robotics technology, path planning and optimization have become fundamental areas of research for achieving efficient and safe autonomous robot navigation. In this paper, we review the classic and state-of-the-art techniques of path planning and optimization, including artificial potential fields, A* algorithm, Dijkstra's algorithm, genetic algorithm, swarm intelligence, and machine learning-based methods. We analyze the strengths and weaknesses of each approach and discuss their application scenarios. Moreover, we identify the challenges and open problems in this field, such as dealing with dynamic environments and real-time constraints. This paper serves as a comprehensive reference for researchers and practitioners in the robotics community, providing insights into the latest trends and developments in path planning and optimization for mobile robots.