abstract: In the last two decades a major progress has been made in volcanology due to the introduction of mathematical models in the study of volcanic processes. First, by using simplified steady-state 1D models, main mechanisms and controlling processes were recognized in the dynamics of volcanic eruptions. More recently, the development of fully transient, 3D and multiphase flow models able to solve the transport equations of the eruptive mixture, allowed to make a further progress in the description of the natural system. In this talk a few examples of mathematical models used in the numerical simulation of eruptive processes will be presented with specific application to the Italian explosive volcanoes.