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A wide range of human–robot collaborative applications in diverse domains, such as manufacturing, health care, the entertainment industry, and social interactions, require an autonomous robot to follow its human companion. Different working environments and applications pose diverse challenges by adding constraints on the choice of sensors, degree of autonomy, and dynamics of a person-following robot. Researchers have addressed these challenges in many ways and contributed to the development of a large body of literature. This paper provides a comprehensive overview of the literature by categorizing different aspects of person-following by autonomous robots. Also, the corresponding operational challenges are identified based on various design choices for ground, underwater, and aerial scenarios. In addition, state-of-the-art methods for perception, planning, control, and interaction are elaborately discussed and their applicability in varied operational scenarios is presented. Then some of the prominent methods are qualitatively compared, corresponding practicalities are illustrated, and their feasibility is analyzed for various use cases. Furthermore, several prospective application areas are identified, and open problems are highlighted for future research.