Abstract. We describe a series of models which illustrate the controls upon the evolution of an erupting mixture of ash and gas during an explosive volcanic eruption. For large eruption rates, material typically issues from a crater as a supersonic jet which may entrain and heat sufficient air to become buoyant and form a Plinian eruption column. If a buoyant eruption column is able to form, then this column may ascend to heights of order 10-30 km, depending upon the erupted mass flux. In contrast, for low eruption rates, a shock forms in the crater and the material issues as a slow subsonic flow which generates dense hot ash flows. A new model shows that as such ash flows propagate from the vent, the density of the flow decreases mainly due to sedimentation, until ultimately the residual ash flow becomes buoyant. The distance the flow travels before becoming buoyant increases with the mass flux in the current and the mean size of particles in the current, but decreases with the flow temperature. It also depends upon the mass of air entrained into the collapsing fountain. The mass fraction of solid lifted from such ash flows into the ascending cloud depends mainly upon the mass of air entrained into the collapsing fountain near the volcanic vent. We apply our models to predict run-out distances and deposition patterns produced by erupting volcanoes.