This chapter is about classes of statistical distributions that deliver extreme events, and how we should deal with them for both statistical inference and decision making. It draws on the author’s multi-volume series, Incerto, and associated technical research program, which is about how to live in a real world with a structure of uncertainty that is too complicated for us. The Incerto tries to connect five different fields related to tail probabilities and extremes: mathematics, philosophy, social science, contract theory, decision theory, and the real world. The main idea behind the project is that while there is a lot of uncertainty and opacity about the world, and incompleteness of information and understanding, there is little, if any, uncertainty about what actions should be taken based on such incompleteness, in a given situation.