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Estimating and reacting to disturbances is crucial for robust flight control of quadrotors. Existing estimators typically require significant tuning for a specific flight scenario or training with extensive ground-truth disturbance data to achieve satisfactory performance. In this paper, we propose a neural moving horizon estimator (NeuroMHE) that can automatically tune its key parameters modeled by a neural network and adapt to different flight scenarios. We achieve this by deriving the analytical gradients of the MHE estimates with respect to the MHE weighting matrices, which enables a seamless embedding of the MHE as a learnable layer into the neural network for highly effective learning. Interestingly, we show that the gradients can be computed efficiently using a Kalman filter in a recursive form. Moreover, we develop a model-based policy gradient algorithm to train NeuroMHE directly from the quadrotor trajectory tracking error without needing the ground-truth disturbance data. The effectiveness of NeuroMHE is verified extensively via both simulations and physical experiments on quadrotors in various challenging flights. Notably, NeuroMHE outperforms a state-of-the-art neural network-based estimator, reducing force estimation errors by up to 76.7%, while using a portable neural network that has only 7.7% of the learnable parameters of the latter. The proposed method is general and can be applied to robust adaptive control of other robotic systems.