The bicycle was invented more than two centuries ago. This seemingly ancient vehicle not only provides convenient transportation for people, but also attracts the interest of scholars from mathematics, physics, mechanics and other disciplines due to its unique motion characteristics and dynamic properties. Generally, a bicycle can be described as a rigid multi-body system with seven degrees of freedom, subjected to four nonholonomic constraints. However, due to the complex kinematic coupling between the front and rear wheels of a bicycle, its constraint equations and dynamic model become extremely complicated, leading to some vague knowledge about bicycle self-stability. Aiming at the classic Carvallo-Whipple bike configuration, this paper systematically reviewed the relevant problems in the research of bicycle dynamics in history. These problems include: (1) Mathematical description of geometric constraints and nonholonomic constraints for a bicycle moving on a complex curved surface; (2) The intrinsic symmetries of the bicycle system and the relevant conservation quantities; (3) Various modeling methods of bicycle dynamics; (4) The stability analysis of the relative equilibriums for the bicycle motions in a uniform linear motion on a horizontal surface and in a uniform circular motion on a surface of revolution, respectively; (5) Structural parameters affecting the bicycle self-stability, and etc. Finally, some typical experiments work and the different control strategies of the bicycle are detailedly described, and also several open problems are addressed for future research.