Collaborative and companion robots are at the forefront of technological innovation, transforming human-robot interaction to address a wide range of tasks and activities. This review provides a comprehensive examination of the current state of research on human following and guidance with autonomous mobile robots. Covering the evolution of research from the inception to the latest advancements in this area, we categorize existing literature based on various attributes including fields of application, technologies employed, and social acceptability. We critically analyze and compare state-of-the-art approaches in perception, tracking, planning, control, and human-robot interaction, highlighting their effectiveness and feasibility. We further classify studies based on application domains where person following and guiding tasks are particularly impactful, such as healthcare, personal assistance, logistics, and tour guiding. We identify persistent challenges and outline open problems, offering recommendations for future research directions. Our review aims to serve as a foundational reference for researchers and practitioners, fostering continued innovation and development in the deployment of autonomous robots for human following and guidance.