Shape-changing materials open an entirely new solution space for a wide range of disciplines: from architecture that responds to the environment and medical devices that unpack inside the body, to passive sensors and novel robotic actuators. While synthetic shape-changing materials are still in their infancy, studies of biological morphing materials have revealed key paradigms and features which underlie efficient natural shape-change. Here, we review some of these insights and how they have been, or may be, translated to artificial solutions. We focus on soft matter due to its prevalence in nature, compatibility with users and potential for novel design. Initially, we review examples of natural shape-changing materials—skeletal muscle, tendons and plant tissues—and compare with synthetic examples with similar methods of operation. Stimuli to motion are outlined in general principle, with examples of their use and potential in manufactured systems. Anisotropy is identified as a crucial element in directing shape-change to fulfil designed tasks, and some manufacturing routes to its achievement are highlighted. We conclude with potential directions for future work, including the simultaneous development of materials and manufacturing techniques and the hierarchical combination of effects at multiple length scales.