A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. The results of random walk analysis have been applied to computer science, physics, ecology, economics, psychology and a number of other fields as a fundamental Statistical model for Stochastic process in time. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating random walk hypothesis and the financial status of a gambler can all be modeled as random walks. The term random walk was first introduced by Karl Pearson in 1905.

Various different types of random walks are of interest. Often, random walks are assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest. Some random walks are on graph theory, others on the line, in the plane, or in higher dimensions, while some random walks are on group theory. Random walks also vary with regard to the time parameter. Often, the walk is in discrete time, and indexed by the natural numbers, as in X_0,X_1,X_2,\dots. However, some walks take their steps at random times, and in that case the position X_t is defined for the continuum of times t\ge 0. Specific cases or limits of random walks include the drunkard's walk and Lévy flight. Random walks are related to the diffusion models and are a fundamental topic in discussions of Markov processes. Several properties of random walks, including dispersal distributions, first-passage times and encounter rates, have been extensively studied.