Numerical weather prediction (NWP) models remain the cornerstone of atmospheric sciences. Yet, deep learning (DL) is challenging this paradigm by its ability to capture intricate spatio-temporal patterns and deliver ultra-fast predictions. Analogous to the foundation models (e.g., ChatGPT) in natural language processing, foundation models in the weather/climate domain have also been developed. This paper reviews DL and foundation models for weather prediction by highlighting their strengths and limitations. In particular, we carefully examine them from the perspective of their training paradigms: deterministic predictive learning, probabilistic generative learning, and pre-training & fine-tuning. For each paradigm, we summarize the underlying model architectures, training methods, and respective features. To facilitate further study, we provide a curated repository featuring categorized papers, open-source code, and benchmark datasets. Finally, we discuss and suggest potential research directions across new tasks and models in weather data storage and management, and operational deployment, further inspiring innovations in this rapidly evolving field. GitHub: https://github.com/JimengShi/DL-Foundation-Models-Weather.