This project investigates a general method for improving the structure and quality of sequences generated by a recurrent neural network (RNN) using deep reinforcement learning (RL). Our method, which we call Sequence Tutor, allows models to improve sequence quality with RL, while maintaining information originally learned from data, as well as sample diversity. An RNN is first pre-trained on data using maximum likelihood estimation (MLE), and the probability distribution over the next token in the sequence learned by this model is treated as a prior policy. Another RNN is then trained using reinforcement learning (RL) to generate higher-quality outputs that account for domain-specific incentives while retaining proximity to the prior policy of the MLE RNN. To formalize this objective, we derive novel off-policy RL methods for RNNs from KL-control. The effectiveness of the approach is demonstrated on two applications; 1) generating novel musical melodies, and 2) computational molecular generation for drug discovery. For both problems, we show that the proposed method improves the desired properties and structure of the generated sequences, while maintaining information learned from data.