Smart polymers that are capable of controlled shape transformations under external stimuli have attracted significant attention in the recent years due to the resemblance of this behavior to the biological intelligence observed in nature. Moreover, shape-morphing polymer-based materials are great candidates for biomedical applications due to their adaptive properties, controlled shape transformations, and enriched functionality. In this review, we focus on the recent progress in the field of shape-memory and shape-changing polymers, highlighting their most promising applications in the biomedical field. We include a brief theoretical background on the underlying molecular mechanisms, classification of the polymer systems based on their mechanical properties, and the most promising state-of-the-art processing techniques for the fabrication of intelligent polymer-based materials. The aim of this review is to provide a link between the diverse stimuli-responsive polymer systems and their mechanical properties, and to highlight the most appealing up-to-date biomedical applications for shape-morphing systems.