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Complex plasmas are dusty plasmas in which the density and electric charges of the dust grains are sufficiently high to induce long-range grain-grain interactions, as well as strong absorption of charged-plasma components. Together with the sources replenishing the plasma such systems form a highly dissipative thermodynamically open system that exhibits many features of collective behaviour generally found in complex systems. Most notably among them are self-organized patterns such as plasma crystals, plasma clusters, dust stars and further spectacular new structures. Beyond their intrinsic scientific interest, the study of complex plasmas grows in importance in a great variety of fields, ranging from space-plasma sciences to applied fields such as plasma processing, thin-film deposition and even the production of computer chips by plasma etching, in which strongly interacting clouds of complex plasmas can cause major contamination of the final product.