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There are many challenges for robot navigation in densely populated dynamic environments. This paper presents a survey of the path planning methods for robot navigation in dense environments. Particularly, the path planning in the navigation framework of mobile robots is composed of global path planning and local path planning, with regard to the planning scope and the executability. Within this framework, the recent progress of the path planning methods is presented in the paper, while examining their strengths and weaknesses. Notably, the recently developed Velocity Obstacle method and its variants that serve as the local planner are analyzed comprehensively. Moreover, as a model-free method that is widely used in current robot applications, the reinforcement learning-based path planning algorithms are detailed in this paper.