Computer game-playing agents are almost as old as computers themselves, and people have been developing agents since the 1950's. Unfortunately the techniques for game-playing agents have remained basically the same for almost half a century -- an eternity in computer time. Recently developed approaches have shown that it is possible to develop game playing agents with the help of learning algorithms. This study is based on the concept of algorithms that learn how to play board games from zero initial knowledge about playing strategies. A coevolutionary approach, where a neural network is used to assess desirability of leaf nodes in a game tree, and evolutionary algorithms are used to train neural networks in competition, is overviewed. This thesis then presents an alternative approach in which particle swarm optimization (PSO) is used to train the neural networks. Different variations of the PSO are implemented and compared. The results of the PSO approaches are also compared with that of an evolutionary programming approach. The performance of the PSO algorithms is investigated for different values of the PSO control parameters. This study shows that the PSO approach can be applied successfully to train game-playing agents.