Biohybrid robotics takes an engineering approach to the expansion and exploitation of biological behaviors for application to automated tasks. Here we identify the construction of living buildings and infrastructure as a high-potential application domain for biohybrid robotics, and review technological advances relevant to its future development. Construction, civil infrastructure maintenance, and building occupancy in the last decades have comprised a major portion of economic production, energy consumption, and carbon emissions. Integrating biological organisms into automated construction tasks and permanent building components therefore has high potential for impact. Live materials can provide several advantages over standard synthetic construction materials, including self-repair of damage, increase rather than degradation of structural performance over time, resilience to corrosive environments, support of biodiversity, and mitigation of urban heat islands. Here we review relevant technologies, which are currently disparate. They span robotics, self-organizing systems, artificial life, construction automation, structural engineering, architecture, bioengineering, biomaterials, and molecular and cellular biology. In these disciplines, developments relevant to biohybrid construction and living buildings are in early stages, and typically are not exchanged between disciplines. We therefore consider this review useful to the future development of biohybrid engineering for this highly interdisciplinary application.