In the last decades, volcano monitoring capabilities have increased enormously, thanks to geochemical and geophysical airborne and surface measurements that have steadily improved their accuracy and time resolution. Such a wealth of data is routinely used to track volcanic unrest and eruption evolution, although precise causative links with underground processes are often missing. Modeling of magmatic and volcanic systems has also leaped forward, thanks to the increased availability of computer power and development of numerical models. Capturing the complexity of magmatic system evolution at all scales is nonetheless still a challenge: the crystal- and bubble-size processes need to be taken into account in order to resolve the reservoir-scale dynamics detected by the monitoring networks. We have developed a robust numerical model to solve the thermo-fluid dynamics of magmatic mixtures, that includes pressure-, temperature and melt composition-dependent, locally (space-time) defined properties and constitutive equations of multi-component magmas. The model has been applied to a variety of scenarios related to magma dynamics in underground volcanic systems, including magma arrival from depth into shallow reservoirs. Model results include the space-time evolution of density, pressure, velocity and composition within the domain, that can be used as sources of synthetic geodetic and seismic datasets, akin to those recorded by monitoring networks. The synthetic and monitoring time series can thus be compared and their similarities can be exploited to detect the underground dynamics causing well-defined time and spectral patterns: we are building a physically sound reference for the interpretation of volcanic unrest signals and their relationships with the deep magma dynamics. This approach has been successfully used to detect shallow magma arrival from strainmeter records at Campi Flegrei during the ongoing unrest phase; it is also being applied to evaluate how precise gravity surveys at Mount Etna can help in detecting any shift in the eruptive sequence.