Abstract: Many applications call for estimating probabilities of rare, even previously unseen, events. We outline the problem's theory, applications, and relation to works by Fisher, Shakespeare, Laplace, Good, Turing, Hardy, Ramanujan, and Shannon, and describe recent constructions of asymptotically optimal estimators. The talk is self contained and based on work with P. Santhanam, K. Viswanathan, J. Zhang, and others.