Predicting high impact extreme events, such as severe climatic and economic events is a major societal challenge. Using innovative mathematical techniques this proposal determines phenomenological mechanisms that lead to the occurrence of extremes, and develops a theory that can be used to predict when such events occur in physical modelling applications. Using dynamical systems theory, the proposed research will use geometrical features of the underlying mathematical models to determine the future extreme behaviour. This goes beyond certain traditional approaches such as monitoring output time series data alone. The study of successive maxima (or minima) for stochastic processes is called Extreme Value Theory (EVT). This theory is extensively used in risk analysis to estimate probabilities of rare events and extremes, e.g. high river levels; hurricanes and market crashes. 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 and the time series of observations can be highly correlated. A key question that we address is when to modify the theory for independent, identically distributed 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? Ergodic theory approaches have been very successful in understanding the long-term evolution of these systems. Recent approaches have focused on time series observations which have a unique maxima at a distinguished point in phase space, and whose level set geometries coincide with balls in the ambient (usually Euclidean) metric. However extremes of other physically relevant functions (with geometries beyond nested balls), are also important in applications. This includes energy-like functions or wind speed functionals which play a role in measuring the destructiveness of storms. We therefore go beyond existing methodologies and develop a theory of extremes for physically relevant observable functions. We then apply this theory to explicit dynamical systems (both discrete and continuous) motivated by real-world mathematical models such as for the weather and climate.