Animals exploit soft structures to move effectively in complex natural environments. These capabilities have inspired robotic engineers to incorporate soft technologies into their designs. The goal is to endow robots with new, bioinspired capabilities that permit adaptive, flexible interactions with unpredictable environments. Here, we review emerging soft-bodied robotic systems, and in particular recent developments inspired by soft-bodied animals. Incorporating soft technologies can potentially reduce the mechanical and algorithmic complexity involved in robot design. Incorporating soft technologies will also expedite the evolution of robots that can safely interact with humans and natural environments. Finally, soft robotics technology can be combined with tissue engineering to create hybrid systems for medical applications.