Although the Martian atmosphere does not satisfy general requirements for lightning generation, there is a possibility of electrical discharge in the case of strong surface winds and a resulting extremely large dust mass loading in the course of large dust storms occurring on this planet. On Earth, negative potential gradients of many thousands of volts per meter have been measured during dust storms when winds are sufficiently strong. However, owing to a lower pressure in the Martian atmosphere, the required voltage for electrical breakdown is lower than on Earth. After a brief review of the observations concerning Earth's atmosphere and the laboratory experiments performed to understand these phenomena, the results of numerical simulation of the electrification in Martian dust storms are presented. Known characteristics of Martian dust grains and the Martian atmosphere are considered, and the different forces applied to the dust particles are taken into account. The electrostatic potential and the evolution of the charge density are calculated using Poisson's equation and the continuity equation. Two different wind configurations in the Martian atmosphere are studied: a wind vortex and a wind parallel to the ground encountering an obstacle (a small hill). It is shown that electric discharge occurs under specific conditions.