Extreme space weather events are most dangerous to our society that increasingly depends on space-based assets and the stable availability of electrical power and communication/navigation systems. Learning about such events is, however, difficult because they are also relatively rare. Advancing technologies enable us to obtain the probabilities and properties of extreme geospace weather other than by direct observation. Ice cores contain indirect information on extreme geospace weather in Earth’s past, while observations of stars like the Sun provide a statistical sample of many thousands of stars instead of the single one that we live with. The continuing analysis of multiple ice cores, and the ongoing monitoring of 150,000 stars by the Kepler satellite, for example, provide a wealth of information waiting to be utilized. The ISSI project assembles a team of experts in the analysis of records in ice and rock, solar and stellar observations, and of the processes that link solar events to their observables to (1) set state-of-the-art bounds to the probabilities of the occurrence of space weather events of different magnitudes, and (2) to assess the potential of improving these constraints by identifying gaps in our knowledge or necessary future measurements and techniques to improve our knowledge. The full text of the team proposal to ISSI can be found here.