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In this paper, we present a learning method to solve the vehicle overtaking problem, which demands a multitude of abilities from the agent to tackle multiple criteria. To handle this problem, we propose to adopt a multiple-goal reinforcement learning (MGRL) framework as the basis of our solution. By considering seven different goals, either Q-learning (QL) or double-action QL is employed to determine action decisions based on whether the other vehicles interact with the agent for that particular goal. Furthermore, a fusion function is proposed according to the importance of each goal before arriving to an overall but consistent action decision. This offers a powerful approach for dealing with demanding situations such as overtaking, particularly when a number of other vehicles are within the proximity of the agent and are traveling at different and varying speeds. A large number of overtaking cases have been simulated to demonstrate its effectiveness. From the results, it can be concluded that the proposed method is capable of the following: 1) making correct action decisions for overtaking; 2) avoiding collisions with other vehicles; 3) reaching the target at reasonable time; 4) keeping almost steady speed; and 5) maintaining almost steady heading angle. In addition, it should also be noted that the proposed method performs lane keeping well when not overtaking and lane changing effectively when overtaking is in progress.