Synthetic biology is frequently defined as the application of engineering design principles to biology. Such principles are intended to streamline the practice of biological engineering, to shorten the time required to design, build, and test synthetic gene networks. This streamlining of iterative design cycles can facilitate the future construction of biological systems for a range of applications in the production of fuels, foods, materials, and medicines. The promise of these potential applications as well as the emphasis on design has prompted critical reflection on synthetic biology from design theorists and practicing designers from many fields, who can bring valuable perspectives to the discipline. While interdisciplinary connections between biologists and engineers have built synthetic biology via the science and the technology of biology, interdisciplinary collaboration with artists, designers, and social theorists can provide insight on the connections between technology and society. Such collaborations can open up new avenues and new principles for research and design, as well as shed new light on the challenging context-dependence-both biological and social-that face living technologies at many scales. This review is inspired by the session titled "Design and Synthetic Biology: Connecting People and Technology" at Synthetic Biology 6.0 and covers a range of literature on design practice in synthetic biology and beyond. Critical engagement with how design is used to shape the discipline opens up new possibilities for how we might design the future of synthetic biology.