Imagine a robot that could perceive and manipulate rigid objects as well as a human adult. Would a robot that had such amazing capabilities be able to perform the range of practical manipulation tasks we expect in settings such as the home? Consider that such a robot would still be unable to prepare a meal, do laundry, or make a bed because these tasks involve deformable object manipulation. Unlike in rigid-body manipulation, where methods exist for general-purpose pick-and-place tasks regardless of the size and shape of the object, no such methods exist for a similarly broad and practical class of deformable object manipulation tasks. The problem is indeed challenging, as these objects are not straightforward to model and have infinite-dimensional configuration spaces, making it difficult to apply established motion planning approaches. Our approach seeks to bypass these difficulties by representing deformable objects using simplified geometric models at both the global and local planning levels. Though we cannot predict the state of the object precisely, we nevertheless can perform tasks such as cable-routing, cloth folding, and surgical probe insertion in geometrically-complex environments. Building on this work, our new projects in this area aim to blend exploration of the model space with goal-directed manipulation of deformable objects and to generalize the methods we have developed to motion planning for soft robot arms, where we can exploit contact to mitigate the actuation uncertainty inherent in these systems.