Mathematical models of performance in locomotor sports are reducible to functions of the sorty = f(x)whereyis some performance variable, such as time, distance or speed, andxis a combination of predictor variables which may include expressions for power (or energy) supply and/or demand. The most valid and useful models are first-principles models that equate expressions for power supply and power demand. Power demand in cycling is the sum of the power required to overcome air resistance and rolling resistance, the power required to change the kinetic energy of the system, and the power required to ride up or down a grade. Power supply is drawn from aerobic and anaerobic sources, and modellers must consider not only the rate but also the kinetics and pattern of power supply. The relative contributions of air resistance to total demand, and of aerobic energy to total supply, increase curvilinearly with performance time, while the importance of other factors decreases. Factors such as crosswinds, aerodynamic accessories and drafting can modify the power demand in cycling, while body configuration/orientation and altitude will affect both power demand and power supply, often in opposite directions.Mathematical models have been used to solve specific problems in cycling, such as the chance of success of a breakaway, the optimal altitude for performance, creating a ‘level playing field’ to compare performances for selection purposes, and to quantify, in the common currency of minutes and seconds, the effects on performance of changes in physiological, environmental and equipment variables. The development of crank dynamometers and portable gas-analysis systems, combined with a modelling approach, will in the future provide valuable information on the effect of changes in equipment, configuration and environment on both supply and demand-side variables.