Optical signal-to-noise ratio (OSNR) monitoring is essential to both the operation of reliable and reconfigurable network and the supply of high quality-of-service. Recently, deep learning technique has been proposed for the implementation of OSNR monitoring. However, the potential of deep learning technique for OSNR monitoring has not been fully exploited in terms of the monitoring accuracy and robust operation. Here, we propose an OSNR monitoring scheme with high accuracy and short response time using the long short-term memory neural network (LSTM-NN). The use of LSTM-NN is helpful to identify the relationship between the time-varied data and corresponding OSNR without manual feature extraction. We investigate the optimal number of time steps for the LSTM-NN-based OSNR monitoring and the monitoring accuracy with respect to various modulation formats with variable baud-rates. Finally, we implement an experimental verification for 34.94-GBd polarization division multiplexing-16 quadrature amplitude modulation signal, achieving a mean absolute error of 0.05 dB over the OSNR range from 15 to 25 dB. The advantages of LSTM-NN-based OSNR monitoring include high accuracy and smart response, which is ideal for future agile optical network.