Harvesting robots represent a critical enabling technology for advancing the mechanization and automation of fruit and vegetable harvesting and have become a prominent research focus within the global agricultural robotics community. Unlike structured industrial environments, agricultural harvesting tasks are performed in highly unstructured and dynamic settings, where fruits, branches, and leaves are densely intertwined and frequently occluded one another. As a result, obstacle-aware operation capability has emerged as a key technological bottleneck that fundamentally limits the overall performance, robustness, and practical applicability of fruit and vegetable harvesting robots. In this context, a comprehensive review of recent advances in obstacle-avoidance technologies for harvesting robots operating in complex agricultural environments was provided. The review was structured around four core aspects that were critical to obstacle-aware harvesting: target perception, manipulation decision-making, servo control, and end-effector and execution structures. Firstly, advances in visual and multimodal perception methods for detecting and segmenting fruits, branches, and foliage under occlusion were examined, with particular attention paid to deep learning-based approaches and three-dimensional sensing techniques. Secondly, manipulation decision-making strategies, including motion planning, behavior selection, and learning-based decision models, were reviewed with respect to their ability to cope with high-dimensional constraints and environmental uncertainty. Thirdly, servo control methods for harvesting robots were discussed, focusing on visual servoing, force-aware control, and adaptive strategies that enabled precise and safe manipulation in cluttered scenes. Finally, the design of execution mechanisms and end-effectors was analyzed, highlighting how mechanical structure, compliance, and functional integration influence obstacle avoidance performance during harvesting operations. Based on this review, it was further analyzed and summarized the major technical challenges faced by obstacle-aware harvesting robots in real-world agricultural scenarios. These challenges included limited perception reliability under severe occlusion, insufficient generalization of decision and control strategies across varying crop types and growth stages, difficulties in reproducing human harvesting skills, and the lack of coordination between robotic system design and agricultural production practices. Finally, future research trends were discussed, emphasizing the potential of embodied intelligence, end-to-end learning frameworks, and the integration of agronomic knowledge with robotic design. With advancements in factory-based agronomic management, artificial intelligence, and robotic technologies, the development of embodied intelligence—particularly supported by multimodal information perception and environmental interactive learning, and backed by China’s intelligent robotics industry would serve as a crucial technical pathway for enhancing the capability of agricultural robots in handling complex operational tasks.