Robot person following (RPF) -- mobile robots that follow and assist a specific person -- has emerging applications in personal assistance, security patrols, eldercare, and logistics. To be effective, such robots must follow the target while ensuring safety and comfort for both the target and surrounding people. In this work, we present the first comprehensive study of RPF, which (i) surveys representative scenarios, motion-planning methods, and evaluation metrics with a focus on safety and comfort; (ii) introduces Follow-Bench, a unified benchmark simulating diverse scenarios, including various target trajectory patterns, crowd dynamics, and environmental layouts; and (iii) re-implements six representative RPF planners, ensuring that both safety and comfort are systematically considered. Moreover, we evaluate the two best-performing planners from our benchmark on a differential-drive robot to provide insights into the real-world deployment of RPF planners. Extensive simulation and real-world experiments provide a quantitative study of the safety-comfort trade-offs of existing planners, while revealing open challenges and future research directions.