Genetic Algorithms (GAs) are a class of numerical and combinatorial optimisers which are especially useful for solving non-linear and non-convex problems. However, the required execution time often limits their application to small-scale or latency-insensitive problems, so techniques to increase the speed and computational efficiency of GAs are needed. FPGA-based acceleration has significant potential for speeding up genetic algorithms, but existing FPGA-based GA systems are limited by the need for extensive hardware architecture customisation by end users and the relatively small improvement in performance. The main reasons for these limitations are that most of the units in these GA systems are fixed and the algorithms of these FPGA-based systems are just inherited from software GAs without making full use of reconfigurable hardware. This thesis examines the previous work and then rethinks the creation of genetic algorithms with reconfigurable computing, and aims to address the problems of low-level programmability, flexibility, and performance in current GA systems. The thesis first presents a general-purpose automated framework for creating and executing a flexible parallel GA system on one FPGA or even multiple FPGAs, which increases the flexibility in GA architecture and removes the low-level hardware programmability barrier for non-expert users. The thesis then proposes a novel hardware-oriented approach called Pipelined Genetic Propagation (PGP), which is intrinsically spatially distributed and fully pipelined. PGP represents a GA solver as a graph of loosely coupled genetic operators, which allows the system to be scaled to the available resources, and also to dynamically change connection topology at run-time to explore different strategies. Finally the thesis extends PGP in three ways: Recursive PGP for multiple level optimisation problems, machine learning guided PGP for automatically changing the connection topology, and partially reconfigurable PGP to increase flexibility. Overall, the thesis presents new ways to perform genetic algorithms in reconfigurable computing, with high-level programmability, sufficient flexibility and high performance.