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Volcano deformation monitoring is one of the most widely studied research topics by using interferometric synthetic aperture radar (InSAR). In the new open and big SAR data era, the development of data processing methods, and interdisciplinary integration have pushed volcanic study into a new stage. This paper reviews the studies of volcano monitoring and parameter inversion based on InSAR. First, we introduce the principle of InSAR, the advanced InSAR time series methods, and the main error sources. Then, we summarize the commonly used classical analytical and numerical models in volcanic deformation modeling. We select several cases to introduce the specific InSAR applications in studying volcanic tectonic dynamic mechanisms including magma transfer, eruption, degassing and hydrothermal activity. Finally, we summarize the current status of volcanic deformation monitoring in China and the future advances of InSAR volcano geodesy. [ABSTRACT FROM AUTHOR]