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This paper presents a review of the design parameters, mathematical techniques and simulations used in the design of parabolic trough solar systems, along with a review on their applications. Recent studies that analyze the deployment of solar parabolic trough collectors (SPTC) in different countries and the operational SPTC plants are also presented and discussed. The paper also discusses the different kinds of software and test methods of solar collectors developed since 1981 which can be distinguished by their particular mathematical models or tracking techniques. In particular, since the mathematical models are especially required for the design, analysis, testing and validation of the systems results as they provide an approximation of the dynamic behavior of the physical properties of the system, they are discussed in depth. The mathematical models allow the calculation of different parameters of the solar parabolic trough system, the angle of inclination of the collecting surface and the forces acting on the system. The validity and experimental validation of the major mathematical models on practical solar parabolic trough concentrators, receivers and other components of different dimension are also reviewed. The paper showed the optical efficiency values are close to 63% and the theoretical peak optical efficiency reached 75%. [ABSTRACT FROM AUTHOR]