The concept of a field dates back to Euler, who introduced the notion to describe fluid flows in his study of hydrodynamics. Methods and concepts based on field theory now pervade the physical sciences and engineering. Field-theoretical ideas exert a strong influence on physical intuition and shape modern nomenclature. In addition, some of the most powerful analytical tools available to the modern scientist are those developed to study the behavior of fields.

In the context of models designed to describe the physical world, a field is a quantity that varies continuously in space and time. Examples are the electric and magnetic fields, the velocity and density distributions of a liquid or vapor and the quantum-mechanical wavefunction of a microscopic particle. In some cases, such as the velocity and density fields introduced by Euler, the notion of continuity must be taken advisedly. Because of the atomic structure of matter, one cannot carry the notion of a smooth density distribution down to the length scales on which molecules can be distinguished. There, the classical description is necessarily in terms of particles. Quantum mechanically, wavefunctions replace the classical density and velocity fields as the appropriate mode of description. This proviso notwithstanding, in the regimes in which density and velocity fields accurately describe the state of a liquid or vapor, they form the basis of an extremely useful theoretical model that yields important physical properties of these systems.

It turns out that the random walk also lends itself to description in terms of a field. As in the case of a liquid or vapor, the field-based description maintains its validity in a restricted range of length scales.