The violent nature of explosive volcanic eruptions makes understanding their behavior both imperative and extremely challenging. These dangerous natural phenomena threaten society in a variety of ways ranging from destruction of local communities to disrupting global air traffic to influencing global climate change. Our ability to mitigate the risks posed by volcanoes is hampered by our limited understanding of their controlling physics. The opacity and violence of eruptions makes them difficult and dangerous to measure directly. We therefore depend strongly on numerical models to study and understand eruptive dynamics. These models range in sophistication and computational expense depending on their purpose. Real-time hazard assessment requires parameterized models that can run quickly on a desktop computer during eruptions. On the other hand, 3D time-dependent computational fluid dynamics models run on massively parallel supercomputers are necessary to capture and study the turbulent, multiphase flow that controls eruption behavior. This talk will discuss some of the computational issues and challenges currently faced by the volcanology community.