This paper reviews the robotic manipulation of deformable objects in caregiving scenarios. Deformable objects like clothing, food, and medical supplies are ubiquitous in care tasks, yet pose modeling, control, and sensing challenges. This paper categorises caregiving deformable objects and analyses their distinct properties influencing manipulation. Key sections examine progress in simulation, perception, planning, control, and system designs for deformable object manipulation, along with end-to-end deep learning’s potential. Hybrid analytical data-driven modeling shows promise. While laboratory successes have been achieved, real-world caregiving applications lag behind. Enhancing safety, speed, generalisation, and human compatibility is crucial for adoption. The review synthesises critical technologies, capabilities, and limitations, while also pointing to open challenges in deformable object manipulation for robotic caregiving. It provides a comprehensive reference for researchers tackling this socially valuable domain. In conclusion, multi-disciplinary innovations combining analytical and data-driven methods are needed to advance real-world robot performance and safety in deformable object manipulation for patient care.