The Southern Volcanic Zone (SVZ) is one of the most active volcanic provinces of South America, having an unprecedented large amount of eruptions between 2008 and 2018. In this thesis, I use Interferometric Synthetic Aperture Radar (InSAR) geodesy to constrain mechanisms of magma storage, transfer and eruption in a wide range of eruptions in the SVZ spanning from small Strombolian (VEI 1-2) to large Plinian (VEI 4-5). In general, ground deformation of 101-102 cm is only recorded during VEI 4-5 eruptions, while the VEI 1-2 events tend to lack signals of this magnitude. Eruptions in the SVZ are triggered by transient pulses of magma injection, that last 10-1–101, years into shallow sill-like reservoirs rather than by volatile exsolution. Magma is then transported from these shallow reservoirs to the eruptive vents by conduits rather than dikes. Aside from the 2008-2009 Chaiten eruption, there is no conclusive geodetic evidence in the SVZ for dike intrusions. In terms of specific volcanoes, I detect precursory inflation for at least 6 years before the Hudson volcano 2011 eruption. I also detect ground deformation during different stages of the eruptive cycle at Villarrica, Llaima, and Calbuco volcanoes, although there is very little evidence of pre-eruptive geodetic signals for these three edifices. Several mechanisms including open conduits, spatiotemporal aliasing and signals below the detection threshold can explain the overall lack of ground deformation. On the other hand, InSAR records an eleven-year cycle of three pulses of pre-eruptive uplift, co-eruptive deflation, and three different pulses of post-eruptive magma recharge at Cordon Caulle volcano. The sources responsible for these transient pulses of deformation are shallow, at depths between 4 and 6 km, and located in different parts of the volcano. I suggest that these magma injection pulses remobilize the melt in the crystal mush that likely underlies the edifice. Further, a unique set of InSAR ground deformation and topographic data recorded during the VEI 4-5 eruption of this volcano between June 2011 and March 2012 provides the first quantitative constraints on the evolution of rhyolitic lava extrusion and the deflation of the reservoir that stored this magma. A physicochemical model forecasts the quasi-exponential trend of ground deflation and lava effusion, and predicts that the reservoir had an H2O content halfway between a gas-rich and a completely degassed rhyolite, consistent with the observed hybrid explosive - effusive eruptive style.