Volcanic eruptions are amongst the most spectacular of natural phenomena. During the past few years, our knowledge of the basic physics and chemistry of magmatic and volcanic processes has been transformed. We now understand how rocks melt and how the resulting magmas migrate upward and eventually accumulate to form magma chambers. We can model a range of physical and chemical processes in these chambers, and we recognize why chamber walls fracture and how magma flows away from chambers through fractures in the crust. Experimental and theoretical work on bubble nucleation and growth has improved models of magma ascent and explosive volcanic eruption mechanisms. Detailed geophysical, geochemical and field research has led to major advances in understanding the mechanisms that trigger changes between lava-forming (or effusive) eruptions and explosive eruptions. Numerical models that simulate processes involved in explosive and effusive eruptions are at an advanced stage but are capable of further refinement. Many of these advances have been made through interdisciplinary collaboration between geologists and colleagues in other areas, especially physics and mathematics. In this contribution, we review the results of some of this work, and draw attention to areas where future collaboration will help to unravel outstanding problems in volcanology.