Nature's material systems during evolution have developed the ability to respond and adapt to environmental stimuli through the generation of complex structures capable of varying their functions across direction, distances and time. 3D printing technologies can recapitulate structural motifs present in natural materials, and efforts are currently being made on the technological side to improve printing resolution, shape fidelity, and printing speed. However, an intrinsic limitation of this technology is that printed objects are static and thus inadequate to dynamically reshape when subjected to external stimuli. In recent years, this issue has been addressed with the design and precise deployment of smart materials that can undergo a programmed morphing in response to a stimulus. The term 4D printing was coined to indicate the combined use of additive manufacturing, smart materials, and careful design of appropriate geometries. In this review, we report the recent progress in the design and development of smart materials that are actuated by different stimuli and their exploitation within additive manufacturing to produce biomimetic structures with important repercussions in different but interrelated biomedical areas.