Given a sequence of random variables X_1, X_2, X_3,....., extreme statistics is concerned with the limiting distribution of the Maximum: M_n=Max(X_1,...X_n), (under a linear scaling perhaps) as time n goes to infinity.

I shall commence with an overview of extreme value theory in the context of weakly dependent random variables. With examples, I will then describe recent developments in the context deterministic (but chaotic) dynamical systems. Particular examples include non-uniformly expanding maps, Gibbs Markov maps, Young towers, and hyperbolic flows. This is work in progress and is joint with M. Nicol and A. Torok (University of Houston).