Soft manipulator is a kind of special manipulator that uses soft material or flexible structure to perform manipulation task under the specific drive mode inspired by the soft tissue. It is developing gradually as an important component of the soft robot. In this paper, the typical design cases of the soft manipulators as well as the related research teams are firstly investigated. The characteristics of different drive modes are compared and analyzed. Based on these investigations, the enabling technologies are systematically summarized in five aspects, such as kinematic and dynamic modeling, motion control, shape detection, dynamic tactile perception, and stiffening method. Currently, the research on soft manipulator has initially formed a technical system, although it is still in the primary stage. With the development of the soft manipulator, its overall technology maturity will be gradually improved. Currently, the applications of the soft manipulator lie mainly in the fields of the ground rescue, the underwater grabbing, the medical operation, the assist maintenance, the man-machine interaction, the space manipulation and so on. The application fields, such as origami soft manipulator, biomimetic soft microrobots, self-healing soft robot, space soft manipulator and so on, emerge new opportunities and challenges for the soft manipulators.