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Explore the advancements in modeling explosive volcanic eruptions and their associated hazards in this comprehensive AGU conference talk. Delve into the critical importance of understanding volcano dynamics and forecasting hazards, given that approximately 10% of the world's population lives within 100 km of historically active volcanoes. Examine the evolution of physical-mathematical models, from 1D steady-state homogeneous flow models to advanced 2D/3D multiphase flow simulations. Discover how these models have improved our understanding of complex volcanic processes, including conduit flow, convective plumes, and pyroclastic density currents. Learn about the power of 3D modeling with DEM topography in quantifying potential eruptive scenarios and associated hazards. Investigate the use of simplified flow models combined with Monte Carlo and statistical methods to produce probabilistic hazard maps. Gain insights into specific case studies of high-risk volcanoes in Italy, the Caribbean, and the United States. Conclude by exploring the future challenges in modeling research and its potential contributions to mitigating volcanic emergencies.