In many seakeeping applications, it is desirable to know the probability that a particular random process, such as the height of the sea at a given point, exceeds a predetermined level during an interval of time. The basic mathematical analysis of extreme values is presented. Bounds are obtained for two distinct cases: In the first case, only the covariance function of the process is assumed to be known; while in the second case, the joint distribution of the process and its first derivative are considered known. The results are applied to various seakeeping applications. In particular, bounds on the probability of extreme wave heights are developed and analyzed using the proposed spectrum of Pierson and Moskowitz for fully developed seas. Similar results are also obtained for the energy associated with a random process. Finally, recommendations are made to extend the work on both a theoretical and experimental basis.