The volume and variety of Earth data have increased as a result of growing attention to climate change and, subsequently, the availability of large-scale sensor networks and remote sensing instruments. This data has been an important resource for data-driven studies to generate practical knowledge and services, support environmental modeling and forecasting needs, and transform climate and earth science research thanks to the increased availability of computational resources and the popularity of novel computational techniques like deep learning. Timely and accurate simulation and modeling of extreme events are critical for planning and mitigation in hydrology and water resources. There is a strong need for short-term and long-term forecasts of streamflow, benefiting from recent developments in data availability and novel deep learning methods. In this study, we review the literature for studies that employ deep learning in tackling tasks that are either to improve the quality of the streamflow data or to forecast streamflow. The study aims to serve as a starting point by covering the latest developments of deep learning approaches in those topics as well as highlighting problems, limitations, and open questions with insights for future directions.