In recent years, soft robotics technologies enabled the development of a new generation of biomedical devices. The combination of elastomeric materials with tunable properties and muscle-like motions paved the way toward more realistic phantoms and innovative soft active implants as artificial organs or assistive mechanisms. This review collects the most relevant studies in the field, giving some insights about their distribution in the past 10 years, their level of development and opening a discussion about the most commonly employed materials and actuating technologies. The reported results show some promising trends, highlighting that the soft robotics approach can help replicate specific material characteristics in the case of static or passive organs but also reproduce peculiar natural motion patterns for the realization of dynamic phantoms or implants. At the same time, some important challenges still need to be addressed. However, by joining forces with other research fields and disciplines, it will be possible to get one step closer to the development of complex, active, self-sensing and deformable structures able to replicate as closely as possible the typical properties and functionalities of our natural body organs.