Professional race-car drivers can execute extreme overtaking maneuvers. However, existing algorithms for autonomous overtaking either rely on simplified assumptions about the vehicle dynamics or try to solve expensive trajectory-optimization problems online. When the vehicle approaches its physical limits, existing model-based controllers struggle to handle highly nonlinear dynamics, and cannot leverage the large volume of data generated by simulation or real-world driving. To circumvent these limitations, we propose a new learning-based method to tackle the autonomous overtaking problem. We evaluate our approach in the popular car racing game Gran Turismo Sport, which is known for its detailed modeling of various cars and tracks. By leveraging curriculum learning, our approach leads to faster convergence as well as increased performance compared to vanilla reinforcement learning. As a result, the trained controller outperforms the built-in model-based game AI and achieves comparable overtaking performance with an experienced human driver.