Tracking and predicting extreme events in large-scale spatio-temporal climate data are long standing challenges in climate science. In this paper, we propose Convolutional LSTM (ConvLSTM)-based spatio-temporal models to track and predict hurricane trajectories from large-scale climate data; namely, pixel-level spatio-temporal history of tropical cyclones. To address the tracking problem, we model time-sequential density maps of hurricane trajectories, enabling to capture not only the temporal dynamics but also spatial distribution of the trajectories. Furthermore, we introduce a new trajectory prediction approach as a problem of sequential forecasting from past to future hurricane density map sequences. Extensive experiment on actual 20 years record shows that our ConvLSTM-based tracking model significantly outperforms existing approaches, and that the proposed forecasting model achieves successful mapping from predicted density map to ground truth.