The aging of buildings is a global concern, with potential risks to human safety and property. Building inspection and maintenance are crucial for ensuring structural integrity and safety. However, traditional manual methods are time-consuming and pose safety risks, especially for exterior inspection at height. Robotics offer a promising alternative to enhance building inspection efficiency and cost-effectiveness, but still in the early development stage. This paper aims to review and analyze the state-of-the-art design and development of robotics for building exterior inspection, referring to the literature published in the last two decades. Firstly, the review classifies different types of robots for building exterior inspection in terms of locomotion and adhesion modes, and discusses the capability of robots from navigation, obstacle surmounting, wall-to-wall/floor transition, curved wall climbing, grasping, barrier avoidance, and self-protection. Secondly, the paper examines the applicability of robots to various building materials for inspections and summarizes the most typical applications (i.e. glass curtain walls, tile walls, and concrete walls). Thirdly, the paper discusses the typical data collection and analysis methods for building exterior inspection using robots. The paper also explored potential enhancements for robotic inspection through the integration of building information modeling, augmented reality/virtual reality, and the involvement of human-in-the-loop. Finally, the paper summarizes the typical application of robotics in building exterior inspection regarding robot types, inspection applications, data collection and analysis methods, discusses the challenges, and outlines the future directions.