Atmospheric extreme events cause severe damage to human societies and ecosystems. The frequency and intensity of extremes and other associated events are continuously increasing due to climate change and global warming. The accurate prediction, characterization, and attribution of atmospheric extreme events is, therefore, a key research field in which many groups are currently working by applying different methodologies and computational tools. Machine learning and deep learning methods have arisen in the last years as powerful techniques to tackle many of the problems related to atmospheric extreme events. This paper reviews machine learning and deep learning approaches applied to the analysis, characterization, prediction, and attribution of the most important atmospheric extremes. A summary of the most used machine learning and deep learning techniques in this area, and a comprehensive critical review of literature related to ML in EEs, are provided. The critical literature review has been extended to extreme events related to rainfall and floods, heatwaves and extreme temperatures, droughts, severe weather events and fog, and low-visibility episodes. A case study focused on the analysis of extreme atmospheric temperature prediction with ML and DL techniques is also presented in the paper. Conclusions, perspectives, and outlooks on the field are finally drawn.