Advances in soft robotics will be crucial to the next generation of robot–human interfaces. Soft material systems embed safety at the material level, providing additional safeguards that will expedite their placement alongside humans and other biological systems. However, in order to function in unpredictable, uncontrolled environments alongside biological systems, soft robotic systems should be as robust in their ability to recover from damage as their biological counterparts. There exists a great deal of work on self-healing materials, particularly polymeric and elastomeric materials that can self-heal through a wide variety of tools and techniques. Fortunately, most emerging soft robotic systems are constructed from polymeric or elastomeric materials, so this work can be of immediate benefit to the soft robotics community. Though the field of soft robotics is still nascent as a whole, self-healing and damage resilient systems are beginning to be incorporated into three key support pillars that are enabling the future of soft robotics: actuators, structures, and sensors. This article reviews the state-of-the-art in damage resilience and self-healing materials and devices as applied to these three pillars. This review also discusses future applications for soft robots that incorporate self-healing capabilities.