This article delivers an extensive review of path planning and obstacle avoidance methods in mobile robotics, covering their theoretical principles, algorithmic progress, and real-world applications. Path planning techniques are grouped into four main categories—classical, sampling-based, optimization-oriented, and learning-based—each discussed in terms of advantages, shortcomings, and suitability for diverse operational contexts. Obstacle avoidance is analyzed through reactive, predictive, and learning-focused approaches, with particular attention to sensor technologies and real-time decision-making. The paper also considers integrated navigation systems that merge global and local planning, utilize layered control structures, and operate on embedded platforms to ensure safe and efficient mobility in complex, dynamic environments. Practical examples, such as the ROS Navigation Stack, autonomous delivery systems, and robotic cleaners, illustrate real-world implementations. Additionally, the article highlights persistent challenges and open research directions, including planning under uncertainty, real-time adaptability, socially aware navigation, coordination of multiple robots, and transfer learning for generalization. The discussion is reinforced with figures and tables comparing algorithmic trade-offs and system designs. Overall, the review provides researchers and practitioners with a structured taxonomy, comparative analysis, and forward-looking perspectives to support the creation of more reliable, adaptive, and intelligent navigation systems for future autonomous robots.