Robot manipulation, for example, pick-and-place manipulation, is broadly used for intelligent manufacturing with industrial robots, ocean engineering with underwater robots, service robots, or even healthcare with medical robots. Most traditional robot manipulations adopt 2-D vision systems with plane hypotheses and can only generate 3-DOF (degrees of freedom) pose accordingly. To mimic human intelligence and endow the robot with more flexible working capabilities, 3-D vision-based robot manipulation has been studied. However, this task is still challenging in the open world especially for general object recognition and pose estimation with occlusion in cluttered backgrounds and human-like flexible manipulation. In this article, we propose a comprehensive analysis of recent progress about the 3-D vision for robot manipulation, including 3-D data acquisition and representation, robot-vision calibration, 3-D object detection/recognition, 6-DOF pose estimation, grasping estimation, and motion planning. We then present some public datasets, evaluation criteria, comparisons, and challenges. Finally, the related application domains of robot manipulation are given, and some future directions and open problems are studied as well.