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Extremal distributions, or extreme value distributions (EVDs), are a class of probability distributions that describe the maximum or minimum of a large number of random variables. They are particularly important in fields such as risk management, environmental science, and engineering, where understanding the probability of rare, extreme events is crucial. The theory of extreme value distributions is a branch of statistics that deals with the statistical behavior of extremes.

There are three main types of extreme value distributions, each corresponding to a different type of limiting behavior of the distribution of the maximum (or minimum) of a sample. These are the Gumbel distribution, the Fréchet distribution, and the Weibull distribution. These three distributions can be unified into a single generalized extreme value distribution (GEV). The GEV distribution has three parameters: location, scale, and shape. The shape parameter dictates which of the three basic types the distribution belongs to.

In practice, the extreme value theory is used to model phenomena such as the maximum flood levels in a river, the highest temperature recorded in a region, the maximum stress a material can withstand, or the largest losses in a financial portfolio. By fitting an appropriate extreme value distribution to historical data, one can estimate the probability of extreme events occurring in the future, such as the return period of a flood of a certain magnitude. This allows for better planning and risk mitigation strategies. The methodology involves methods of moments or maximum likelihood estimation to determine the parameters of the fitted distribution.