Statistical discovery questions in astrophysics and high-energy physics often involve mathematical subtleties that mean standard methods (e.g., chi-square) are inappropriate and can lead to misleading results. At the same time Bayesian and classical statistical techniques can lead researchers to differing conclusions. Moreover modern computational strategies are typically infeasible under extreme discovery criteria (4 sigma or more).  This talk explores the statistical challenges that arise in the quantification of discovery and suggests a strategy that combines Bayesian and classical statistical techniques to tackle these challenges.