Mobile robots, including ground robots, underwater robots, and unmanned aerial vehicles, play an increasingly important role in people’s work and lives. Path planning and obstacle avoidance are the core technologies for achieving autonomy in mobile robots, and they will determine the application prospects of mobile robots. This paper introduces path planning and obstacle avoidance methods for mobile robots to provide a reference for researchers in this field. In addition, it comprehensively summarizes the recent progress and breakthroughs of mobile robots in the field of path planning and discusses future directions worthy of research in this field. We focus on the path planning algorithm of a mobile robot. We divide the path planning methods of mobile robots into the following categories: graph-based search, heuristic intelligence, local obstacle avoidance, artificial intelligence, sampling-based, planner-based, constraint problem satisfaction-based, and other algorithms. In addition, we review a path planning algorithm for multi-robot systems and different robots. We describe the basic principles of each method and highlight the most relevant studies. We also provide an in-depth discussion and comparison of path planning algorithms. Finally, we propose potential research directions in this field that are worth studying in the future.