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In this work a detailed study about the implementation of genetic algorithms (GAs) using parallelism and field programmable gate arrays (FPGAs) is presented. Concretely, we use the traveling salesman problem (TSP) as case study. First at all, the TSP is described as well as the GA used for solving it. Afterwards, we present the hardware implementation of this algorithm. We detail 13 different hardware versions, searching that each new version improves the previous one. Many of these improvements are based on the use of parallelism techniques. Finally, the found results are shown and analysed: hardware/software comparisons, resource use, operation frequency, etc. We conclude indicating the parallelism techniques that obtain better results and stating FPGA implementation is better when the problem size increases or when better solutions (nearer to the optimum) must be found.