This paper aims to provide a systematic review of the published results on design optimization techniques in the area of compliant and soft robots, with a focus on the manufacturing processes, actuation methods and application areas. The goal of this work is to provide a comprehensive view and categorization of recent efforts using optimization for improving the design paradigms of such robot technologies, while focusing on the popular methods of topology optimization and generative design. In addition, this paper provides insights into the technical and technological trends that could potentially steer this area towards widespread adoption in domestic and industrial settings.