Recurrent neural networks (RNNs) are the state of the art in sequence modeling for natural language. However, it remains poorly understood what grammatical characteristics of natural language they implicitly learn and represent as a consequence of optimizing the language modeling objective. Here we deploy the methods of controlled psycholinguistic experimentation to shed light on to what extent RNN behavior reflects incremental syntactic state and grammatical dependency representations known to characterize human linguistic behavior. We broadly test two publicly available long short-term memory (LSTM) English sequence models, and learn and test a new Japanese LSTM. We demonstrate that these models represent and maintain incremental syntactic state, but that they do not always generalize in the same way as humans. Furthermore, none of our models learn the appropriate grammatical dependency configurations licensing reflexive pronouns or negative polarity items.