The car-following (CF) model is the core component for traffic simulations and has been built-in in many production vehicles with Advanced Driving Assistance Systems (ADAS). Research of CF behavior allows us to identify the sources of different macro phenomena induced by the basic process of pairwise vehicle interaction. The CF behavior and control model encompasses various fields, such as traffic engineering, physics, cognitive science, machine learning, and reinforcement learning. This paper provides a comprehensive survey highlighting differences, complementarities, and overlaps among various CF models according to their underlying logic and principles. We reviewed representative algorithms, ranging from the theory-based kinematic models, stimulus-response models, and cruise control models to data-driven Behavior Cloning (BC) and Imitation Learning (IL) and outlined their strengths and limitations. This review categorizes CF models that are conceptualized in varying principles and summarize the vast literature with a holistic framework.