The static properties of tensegrity structures have been widely appreciated in civil engineering as the basis of extremely lightweight yet strong mechanical structures. However, the dynamic properties and their potential utility in the design of robots have been relatively unexplored. This paper introduces robots based on tensegrity structures, which demonstrate that the dynamics of such structures can be utilized for locomotion. Two tensegrity robots are presented: TR3, based on a triangular tensegrity prism with three struts, and TR4, based on a quadrilateral tensegrity prism with four struts. For each of these robots, simulation models are designed, and automatic design of controllers for forward locomotion are performed in simulation using evolutionary algorithms. The evolved controllers are shown to be able to produce static and dynamic gaits in both robots. A real-world tensegrity robot is then developed based on one of the simulation models as a proof of concept. The results demonstrate that tensegrity structures can provide the basis for lightweight, strong, and fault-tolerant robots with a potential for a variety of locomotor gaits.