The Sun sporadically produces energetic eruptive events, which can provide an essential insight into plasma conditions unreachable in laboratories. Moreover, they may affect the Earth’s environment and form important hazards for our technologically dependent society. Such events, while being an object of scientific curiosity in the past, have also become a societally important issue for the modern world. The strength and impacts of such events may vary by many orders of magnitudes, including very strong and extreme ones. Attempts to analyze them were made in the past (e.g., ISSI team 231 in 2011–2012, where some members of the present team participated), but because of the small statistics, it remains unclear just what “extreme events” are, what their occurrence probability is, and whether the Sun has a limit in producing such events – theoretical or statistical extrapolation of the known events led to huge uncertainties. The situation had changed in 2012 with the discovery of a spike in cosmogenic isotope data ca. 774–775 AD (Miyake et al., 2012), which was very likely to have resulted from an extreme solar particle storm (Usoskin et al., 2013; Mekhaldi et al., 2015). Following those work, several similar events have been found in the course of past millennia (e.g., Miyake et al., 2013; O’Hare et al., 2019). Thus, a new paradigm of extreme solar events is being formed, which can be now studied in detail. Also, the paradigm of extreme geomagnetic storms is changing presently – the famous Carrington storm of 1859 (Tsurutani et al., 2003) now appears not to have been a unique extreme one on the centennial scale, as similar events have been found to exist (e.g., Hayakawa et al., 2019). Thus, we are now at a situation where new, earlier unknown types of data appear, leading to a paradigm shift, and a coordinated effort of the research community is needed to analyze them and produce a consistent view and set/re-set the paradigm. The ISSI team format provides a perfect forum for such activity. Here we propose to bring together top world scientists in related fields to attack the problem in a coherent way from different directions and to develop a consensus view on solar extreme events, their physics, occurrence probability and consequences. Different sets of data and models will be used to study such events: decades of direct observations, both ground-based and space-borne; a few hundred years of geomagnetic measurements; a few millennia of historical records; ~10,000 years of terrestrial cosmogenic isotope data; millions of years using cosmogenic isotopes in lunar rocks; and the statistics of thousands of stellar observations during the last decade. All these datasets and methods are presently developed by different groups with little communication between them. We propose that the efforts will be discussed and coordinated towards a common task. Questions to be addressed are: what is an extreme solar event? How strong can it be? How often do they occur? What can be the worst-case scenario for such an event? For that, a combined approach using new proxy data (both measured and archival), precise statistical methods, and state-of-the-art models need to be developed, as extreme events may involve physics going beyond the ‘standard’ approaches and method.