A random walk is a sequence of discrete steps in which each step is randomly taken subject to some set of restrictions in allowed directions and step lengths. Random
walks may be taken along a line, in the plane, in space, or in other specified domains.
Self-avoiding walks are walks (random or otherwise)
in which previous steps may not be taken and/or previous portions of the walk may
not be "crossed."

Random walks have interesting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice.

Physically, random thermal perturbations in a liquid are responsible for a random walk phenomenon known as Brownian motion, and the collisions of molecules in a gas are a random walk responsible for diffusion.