Abstract: This review examines the applications of emerging computing technologies—exemplified by artificial intelligence (AI) and quantum computing—in weather forecasting and climate prediction. It systematically analyzes the utilization of AI in meteorological data processing, diagnostic analysis, and long-term climate prediction. In addition, it elaborates on recent progress in applying quantum computing to accelerate climate model simulations and optimize model parameters. The findings indicate that these advanced computational approaches have markedly enhanced the efficiency, precision, and overall capability of weather and climate prediction systems. However, several challenges persist, including the limited interpretability of AI models and the relatively slow development of quantum computing hardware and algorithms. Looking ahead, continued improvements in the interpretability of AI algorithms and the performance of quantum computing hardware are expected to provide more powerful and reliable support for weather forecasting and climate prediction.