abstract: The study of successive maxima (or minima) for stochastic processes is called Extreme Value Theory. It is extensively used in risk analysis to estimate probabilities of rare events and extremes, e.g. floods; hurricanes; market crashes and general exceedance of thresholds. For physical systems modelled by deterministic dynamical systems, especially chaotic dynamical systems a corresponding theory of extremes is yet to be fully understood. These systems are highly sensitive (in both phase space and parameter space) and the time series of observations can be highly correlated. A key question is when to modify the theory for independent, identically distributed (i.i.d.) random variables in the case of understanding extremes for deterministic systems. Conversely when are certain probabilistic limit laws (such as Poisson laws) a good description of the extreme phenomenon? In this talk we consider such questions, and illustrate with concrete dynamical system applications, e.g. chaotic maps, intermittent systems, and hyperbolic systems.