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An attitude control issue is concerned for a quadrotor with external disturbances in this article. For unknown system dynamics, predictor-based neural networks (NNs) are introduced, where prediction errors, angular velocities, are constructed, instead of tracking errors, for updating NNs' weights. This replacement reduces the occurrence of high-frequency oscillations in NNs' approximation. With this improved NNs, a predictor-based NN disturbance observer is then developed for compensation for external disturbances and NNs' approximation errors, and a normalization learning technique is employed for reduction of the number of learning parameters. A predictor-based neural attitude control strategy is proposed for a quadrotor with external disturbances. Furthermore, measurement noise is taken into account in our predictor-based neural attitude control strategy. The Lyapunov-based stability analysis shows that all closed-loop signals in the designed attitude system are semiglobally bounded. A numerical simulation and a hardware-in-loop experiment as well as outdoor flight verify the effectiveness of the proposed anti-disturbance attitude control strategy.