This book introduces a new way of analyzing, measuring and thinking about mega-risks, a "paradigm shift" that moves from single-solutions to multiple competitive solutions and strategies. "Robust simulation" is a statistical approach that demonstrates future risk through simulation of a suite of possible answers. To arrive at this point, the book systematically walks through the historical statistical methods for evaluating risks. The first chapters deal with three theories of probability and statistics that have been dominant in the 20th century, along with key mathematical issues and dilemmas. The book then introduces "robust simulation" which solves the problem of measuring the stability of simulated losses, incorporates outliers, and simulates future risk through a suite of possible answers and stochastic modeling of unknown variables. This book discusses various analytical methods for utilizing divergent solutions in making pragmatic financial and risk-mitigation decisions. The book emphasizes the importance of flexibility and attempts to demonstrate that alternative credible approaches are helpful and required in understanding a great many phenomena.