Numerous deep learning architectures have been developed to accommodate the diversity of time-series datasets across different domains. In this article, we survey common encoder and decoder designs used in both one-step-ahead and multi-horizon time-series forecasting—describing how temporal information is incorporated into predictions by each model. Next, we highlight recent developments in hybrid deep learning models, which combine well-studied statistical models with neural network components to improve pure methods in either category. Lastly, we outline some ways in which deep learning can also facilitate decision support with time-series data. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.