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The most enduring mysteries in volcanology cannot be tackled by one method alone. Different methods give complementary insights on the same structures and processes. In this dissertation I use InSAR (Interferometric Synthetic Aperture Radar), gravimetry, and rock physics in concert with other data to investigate volcano-tectonic interactions and volcanic structure and life cycles. A major theme of my work is integrating geodetic methods with complementary data to synthesize a self-consistent picture of the volcanic system, including using rock physics to quantitatively link geology to geophysical parameters. The Central Volcanic Zone of the Andes contains informative end-members for studying the extremes of volcanic processes. Sabancaya volcano in Peru experienced some of the strongest earthquakes ever recorded near a volcano without coincident eruption, and Uturuncu volcano in Bolivia exhibits signs of unrest despite having no eruptions in over 250,000 years. At Sabancaya volcano I calculate a 4 year long InSAR time series, combining this with other InSAR time series and seismic, thermal, and geological data to determine that magmatic intrusion in fluid-saturated crust triggered the unusually strong seismicity at this volcano. I then turn to Uturuncu volcano, where I generate an updated density contrast model of the volcano's shallow subsurface. Combining this model with a published resistivity model, I argue that Uturuncu's unrest may be partially due to ongoing ore body formation. I build on this work with a quantitative, rock-physics based joint analysis of density contrast, resistivity, and seismic tomography models of Uturuncu's shallow subsurface. My self-consistent, petrologically informed analysis shows that shallow (