This paper introduces two stochastic processes to model dispersal in biological systems: the position jump process and the velocity jump process. The position jump process involves alternating pauses and jumps, with jump direction and distance determined by an integral operator. Under certain conditions, this process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The velocity jump process involves runs separated by reorientations, with new velocities chosen during reorientations. Under certain assumptions, this process leads to the telegrapher's equation, a damped wave equation. The paper derives expressions for mean squared displacement and other experimentally observable quantities. It also discusses generalizations, including resting times between movements. The paper reviews available data on cell and organism movement and shows how such data can be analyzed within the framework provided. Key words: Dispersal — Cell movement — Random walks — Stochastic processes The paper discusses the standard approach to the space jump process, which is a random walk with no correlation between steps. In a one-dimensional uniform lattice, the probability of a walker being at a site after N steps is given by a binomial distribution. For large N and m << N, this leads to a Gaussian distribution. By taking continuum limits, the probability distribution becomes a diffusion equation. The corresponding stochastic process is called a diffusion process without drift, Brownian motion, or a Wiener process. The paper shows how such processes can be analyzed and generalized.This paper introduces two stochastic processes to model dispersal in biological systems: the position jump process and the velocity jump process. The position jump process involves alternating pauses and jumps, with jump direction and distance determined by an integral operator. Under certain conditions, this process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The velocity jump process involves runs separated by reorientations, with new velocities chosen during reorientations. Under certain assumptions, this process leads to the telegrapher's equation, a damped wave equation. The paper derives expressions for mean squared displacement and other experimentally observable quantities. It also discusses generalizations, including resting times between movements. The paper reviews available data on cell and organism movement and shows how such data can be analyzed within the framework provided. Key words: Dispersal — Cell movement — Random walks — Stochastic processes The paper discusses the standard approach to the space jump process, which is a random walk with no correlation between steps. In a one-dimensional uniform lattice, the probability of a walker being at a site after N steps is given by a binomial distribution. For large N and m << N, this leads to a Gaussian distribution. By taking continuum limits, the probability distribution becomes a diffusion equation. The corresponding stochastic process is called a diffusion process without drift, Brownian motion, or a Wiener process. The paper shows how such processes can be analyzed and generalized.