The understanding of self-organized patterns in spatially extended nonlinear dissipative systems is one of the great challenges in modern natural sciences. Such systems can be investigated in an exemplary manner using planar dc and ac gas-discharge systems. Examples for observed patterns are fronts, stripes, hexagons, targets, spirals, dissipative solitons, fractals and other complex patterns. These patterns are also found in systems that microscopically differ in a fundamental way from plasma systems. It is believed that pattern formation in the above mentioned plasma systems can be described within the scope of the drift-diffusion approximation. In addition, there is strong evidence that in many cases the corresponding equations can be transformed to a set of reaction-diffusion equations. It turn out that plasma specific equations provide insight into the underlying microscopic properties of the plasma involved, whereas reaction-diffusion models offer a convenient background for the identification of relevant pattern forming mechanisms and for the discussion of the experimentally observed universal behaviour.