The ocean is beautiful and important, in many ways a unique subject for scientific study. It demands attention at different scales for different reasons. It piques curiosity. It is a fundamental agent in the evolution of our planet, and of the human race. Its study over centuries has provided the foundations for many scientific fields.

In particular, ocean waves allow direct observation and experimentation at an ideal (O(metre-second))``human scale''; the equations governing the propagation of ocean waves are, to a good approximation, near-linear and soluble in simple situations of interest. Direct comparison of waves ``in vivo'', ``in vitro'' and ``in silico'' is possible with high fidelity, and agreement is good for ``central'' characteristics. In more general situations, quantifying the ocean wave environment becomes more problematic. Direct observation is difficult. Ocean wave characteristics are influenced by changes in atmospheric pressure, sea bottom and proximity to land. Wave fields are directional, and non-stationary in time on different time-scales. For large ocean waves, departure from linearity, the occurrence of transient ``wave-wave'' interactions and wave focussing, make physical modelling less reliable and computationally hard; the physics of breaking waves is not fully understood. Waves much be considered in conjunction with other variation of the ocean surface due to (e.g.) tide and surge. Statistical modelling assumes increasing importance.

The extreme ocean environment is therefore an interesting area for the development of applied extreme value analysis; understanding this environment is important to society given the high proportion of the Earth's population that lives in coastal regions, and those who rely on being able to cross oceans safely, or locate installations in or near to it reliably. A typical extreme value analysis is performed using historical data from measurements and outputs from physical simulators, characterising the environment in some region over some period of time, typically of the order of 30 to 100 years. Inference involves estimating the maximum value that might be observed in the region in a time period considerably longer than that of the historical sample, typically of the order of 1000 or 10000 years (or equivalent extreme quantiles of the distribution of the annual maximum).

This talk will outline statistical challenges in estimating marginal and joint characteristics of extreme ocean environments, such as (a) describing non-stationarity with respect to covariates (e.g wave direction), space and time; (b) describing extremal dependence in space and time; and (c) quantifying uncertainty as comprehensively as feasible.