Manipulating cloth-like deformable objects (CDOs) is a long-standing problem in the robotics community. CDOs are flexible (non-rigid) objects that do not show a detectable level of compression strength while two points on the article are pushed towards each other and include objects such as ropes (1D), fabrics (2D) and bags (3D). In general, CDOs’ many degrees of freedom (DoF) introduce severe self-occlusion and complex state–action dynamics as significant obstacles to perception and manipulation systems. These challenges exacerbate existing issues of modern robotic control methods such as imitation learning (IL) and reinforcement learning (RL). This review focuses on the application details of data-driven control methods on four major task families in this domain: cloth shaping, knot tying/untying, dressing and bag manipulation. Furthermore, we identify specific inductive biases in these four domains that present challenges for more general IL and RL algorithms.