A Lévy flight is a random walk in which the step-lengths have a stable distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.

The term "Lévy flight" was coined by Benoît Mandelbrot, who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution, and Rayleigh flight for when the distribution is a normal distribution (which is not an example of a heavy-tailed probability distribution).

The particular case for which Mandelbrot used the term "Lévy flight" is defined by the survivor function of the distribution of step-sizes, U, being

${\displaystyle \Pr(U>u)={\begin{cases}1&:\ u<1,\\u^{-D}&:\ u\geq 1.\end{cases}}}$

Here D is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution.