Manipulation of deformable objects has given rise to an important set of open problems in the field of robotics. Application areas include robotic surgery, household robotics, manufacturing, logistics, and agriculture, to name a few. Related research problems span modeling and estimation of an object's shape, estimation of an object's material properties, such as elasticity and plasticity, object tracking and state estimation during manipulation, and manipulation planning and control. In this survey article, we start by providing a tutorial on foundational aspects of models of shape and shape dynamics. We then use this as the basis for a review of existing work on learning and estimation of these models and on motion planning and control to achieve desired deformations. We also discuss potential future lines of work.