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Deformable object manipulation (DOM) is a primary bottleneck for the real-world application of autonomous robots, requiring advanced frameworks for sensing, perception, modeling, planning, and control. When fragile objects such as soft tissues or fruits are involved, ensuring safety becomes the paramount concern, fundamentally altering the manipulation problem from one of pure trajectory optimization to one of constrained optimization and real-time adaptive control. Existing DOM methodologies, however, often fall short of addressing fragility constraints as a core design feature, leading to significant gaps in real-time adaptiveness and generalization. This review systematically examines individual components in DOM with a focus on their effectiveness in handling fragile objects. We identified key limitations in current approaches and, based on this analysis, discussed a promising framework that utilizes both low-latency reflexive mechanisms and global optimization to dynamically adapt to specific object instances.