Current powered prosthetic and orthotic devices are mostly based on rigid actuators, which exhibit some intrinsic limitations, such as the reduced number of degrees of freedom, high weight, and limited flexibility. As progress is made in the soft robotics field, artificial muscles arise as potential candidates to replace current rigid actuator technologies. Dielectric elastomer actuators (DEAs) are a very promising class of soft actuator and are attracting attention due to their high energy density, fast response, and high actuation strains, similar or even superior to natural muscles. Such remarkable properties can lead the DEAs to be the next generation of actuators for wearable robots, and overcome the limitations imposed by the rigid actuators currently used. This paper reviews the state of art of the applications of DEAs to prostheses, orthoses, and rehabilitation devices. We analyzed the main configurations that are suitable as artificial muscles for the above-mentioned applications and presented the basic model of a dielectric elastomer transducer. When compared to the properties of natural skeletal muscle, some of these configurations stand out. However, there are still some drawbacks that prevent the large-scale application of DEAs, for instance the high operating voltages, durability issues, and nonlinearities that make them difficult to control. We investigated recent advances in materials, control strategies and fabrication methods that tackle these drawbacks, and they indicate that dielectric elastomers are great candidates to be the next generation of actuators for bionic devices.