Statistical signal processing based speech enhancement methods adopt expert knowledge to design the statistical models and linear filters, which is complementary to the deep neural network (DNN) based methods which are data-driven. In this paper, by using expert knowledge from statistical signal processing for network design and optimization, we extend the conventional Kalman filtering (KF) to the supervised learning scheme, and propose the neural Kalman filtering (NKF) for speech enhancement. Two intermediate clean speech estimates are first produced from recurrent neural networks (RNN) and linear Wiener filtering (WF) separately and are then linearly combined by a learned NKF gain to yield the NKF output. Supervised joint training is applied to NKF to learn to automatically trade-off between the instantaneous linear estimation made by the WF and the long-term non-linear estimation made by the RNN. The NKF method can be seen as using expert knowledge from WF to regularize the RNN estimations to improve its generalization ability to the noise conditions unseen in the training. Experiments in different noisy conditions show that the proposed method outperforms the baseline methods both in terms of objective evaluation metrics and automatic speech recognition (ASR) word error rates (WERs).