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In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems. Attempts to produce a mathematical theory of traffic flow date back to the 1920s, when American Economist Frank Knight first produced an analysis of traffic equilibrium, which was refined into Wardrop's first and second principles of equilibrium in 1952. Nonetheless, even with the advent of significant computer processing power, to date there has been no satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques. These models are then developed into traffic forecasts, and take account of proposed local or major changes, such as increased vehicle use, changes in land use or changes in mode of transport (with people moving from bus to train or car, for example), and to identify areas of congestion where the network needs to be adjusted. Traffic behaves in a complex and nonlinear way, depending on the interactions of a large number of vehicles. Due to the individual reactions of human drivers, vehicles do not interact simply following the laws of mechanics, but rather display cluster formation and shock wave propagation, both forward and backward, depending on vehicle density. Some mathematical models of traffic flow use a vertical queue assumption, in which the vehicles along a congested link do not spill back along the length of the link. In a free-flowing network, traffic flow theory refers to the traffic stream variables of speed, flow, and concentration. These relationships are mainly concerned with uninterrupted traffic flow, primarily found on freeways or expressways.

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Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles. Also known as time-continuous models, all car-following models have in common that they are defined by ordinary differential equations describing the complete dynamics of the vehicles' positions and velocities .