Conventional industrial robots with the rigid actuation technology have made great progress for humans in the fields of automation assembly and manufacturing. With an increasing number of robots needing to interact with humans and unstructured environments, there is a need for soft robots capable of sustaining large deformation while inducing little pressure or damage when maneuvering through confined spaces. The emergence of soft robotics offers the prospect of applying soft actuators as artificial muscles in robots, replacing traditional rigid actuators. Dielectric elastomer actuators (DEAs) are recognized as one of the most promising soft actuation technologies due to the facts that: i) dielectric elastomers are kind of soft, motion-generating materials that resemble natural muscle of humans in terms of force, strain (displacement per unit length or area) and actuation pressure/ density; ii) dielectric elastomers can produce large voltage-induced deformation. In this survey, we first introduce the so-called DEAs emphasizing the key points of working principle, key components and electromechanical modeling approaches. Then, different DEA-driven soft robots, including wearable/humanoid robots, walking/serpentine robots, flying robots and swimming robots, are reviewed. Lastly, we summarize the challenges and opportunities for the further studies in terms of mechanism design, dynamics modeling and autonomous control.