With the rapid development of vision sensing, artificial intelligence, and robotics technology, one of the challenges we face is installing more advanced vision sensors on welding robots to achieve intelligent welding manufacturing and obtain high-quality welding components. Depth perception is one of the bottlenecks in the development of welding sensors. This review provides an assessment of active and passive sensing methods for depth perception and classifies and elaborates on the depth perception mechanisms based on monocular vision, binocular vision, and multi-view vision. It explores the principles and means of using deep learning for depth perception in robotic welding processes. Further, the application of welding robot visual perception in different industrial scenarios is summarized. Finally, the problems and countermeasures of welding robot visual perception technology are analyzed, and developments for the future are proposed. This review has analyzed a total of 2662 articles and cited 152 as references. The potential future research topics are suggested to include deep learning for object detection and recognition, transfer deep learning for welding robot adaptation, developing multi-modal sensor fusion, integrating models and hardware, and performing a comprehensive requirement analysis and system evaluation in collaboration with welding experts to design a multi-modal sensor fusion architecture.