Grasping deformable objects remains a challenging operational task for robots in diverse industrial applications. Different characteristics of deformable objects to be gripped need to be considered in the mechanical design of the gripper. Mechanical grippers often rely on sensors and appropriate control strategies to grasp deformable objects. This study classifies deformable objects, grippers and gripper manufacturers, and their corresponding gripping strategies. In the study of control strategies, model-based algorithm control strategies are often ineffective as often the objects to be gripped are unknown in terms of its rigidity and other morphological characteristics. In contrast, model-free algorithms do not need parametric information of the objects as only input-output signal is required. This allows the model-free controlled grippers adapt to diverse and unstructured environments. Finally, the advantages and disadvantages of current deformable object-grasping techniques are discussed and summarized. The challenges and future directions of robots grasping deformable objects are pointed out.