As climate mitigation and adaptation efforts are being implemented, precise and rapidly responsive weather forecasting with high spatio-temporal resolution has become paramount for optimizing the utilization of renewable energy resources, as well as making informed emergency response to extreme weather events. Conventional numerical weather prediction (NWP), based on solving complex physical equations, provides valuable forecasts for society but is computationally intensive and limited by model approximations. Recently, deep learning has emerged as a powerful complement to NWP, achieving notable gains in forecasting accuracy and computational efficiency. This review traces the evolution from NWP to data-driven deep learning approaches, highlighting the strengths of deep learning in short-term forecasts and its challenges in medium- and long-term forecasts, including model uncertainty and interpretability. It also reviews the potential of large-scale deep learning models in improving forecasting performance.