For damaged buildings that had suffered from a destructive earthquake, timely damage inspection is essential to evaluate their residual seismic safety as well as to assess the long-term maintenance capacity for planning a suitable structural rehabilitation strategy for further service. However, the conventional inspection method by human hands is time-consuming, highly costly, and risky for inspectors in the fieldwork. Thus, in this present study, the current situation for structural damage inspection is summarized, the future potentials of newly developed technology in this field are explored, and the application and usage of robotic technologies with unmanned aerial vehicles (UAVs) and mobile robots that are rapidly developed recently have been reviewed and evaluated. In this work an overview is given of the practical applications of the UAV-and mobile robot-based damage inspection, in which these robotic machines are employed as useful tools to inspect and monitor the damaged condition of infrastructure facilities or buildings by collecting photography data. In addition, a conceptual comparison with the conventional approach is given to allow a better understanding of the advantages and limitations of a robotic technology-aimed damage inspection approach. Also, a discussion on instructions, recommendations, existing challenges, and future directions to address the robotic technology-aimed damage inspection approach into further practice is presented.