Geodetic surveying is a core volcano monitoring technique. Measurements of how the crust deforms can give valuable insight into the mechanisms and processes that drive an eruption, and the way in which they change. Various geodetic observables, including ground deformation and gravity changes, have been recorded on Montserrat throughout the eruption. Instrumentation and surveying networks used to make such measurements have evolved significantly since 1995, providing increasingly accurate and robust observations. The detailed research that has been facilitated by these rich geodetic datasets has illuminated many aspects of the Soufrière Hills Volcano (SHV) and demonstrated eruptive mechanisms that are relevant to the study of other volcanoes. We have compiled a history of the geodetic study of the eruption on Montserrat, detailing the development of surveying techniques, network design and data processing since 1995. We then underline some of the key geodetic observations and review some of the most significant research that has contributed to our understanding of this volcanic system. Finally, we apply a series of typical deformation inversion models to deformation observations, and discuss the parameter sensitivity of such modelling approaches and how confidently they can be applied to identify the characteristics of the mechanisms feeding the eruption.