At its core, quantum mechanics is a theory developed to describe fundamental observations in the spectroscopy of solids and gases. Despite these practical roots, however, quantum theory is infamous for being highly counterintuitive, largely due to its intrinsically probabilistic nature. Neural networks have recently emerged as a powerful tool that can extract nontrivial correlations in vast datasets. These networks routinely outperform state-of-the-art techniques in language translation, medical diagnosis, and image recognition. It remains to be seen if neural networks can be trained to predict stochastic quantum evolution without a priori specifying the rules of quantum theory. Here, we demonstrate that a recurrent neural network can be trained in real time to infer the individual quantum trajectories associated with the evolution of a superconducting qubit under unitary evolution, decoherence, and continuous measurement from physical observations only. The network extracts the system Hamiltonian, measurement operators, and physical parameters. It is also able to perform tomography of an unknown initial state without any prior calibration. This method has the potential to greatly simplify and enhance tasks in quantum systems such as noise characterization, parameter estimation, feedback, and optimization of quantum control.