Point process theory proves invaluable across various domains, including insurance, finance, biology, meteorology, genetics and seismology, offering insights and solutions to a myriad of challenges. Among these, Poisson processes stand out as widely applied, serving as fundamental components in numerous point process models, notably the Cram´er–Lundberg model. This talk centers on specialized point process models designed to account for clustering (for example in insurance events), replacing the conventional Cram´er–Lundberg model with a more adept framework capable of capturing such clusters. The focus lies on describing the limiting distribution of the extremes of observations which arrive in clusters. The talk begins with an examination of the tail behavior within a single cluster. Following this groundwork, the developed result is then applied to determine the limiting distribution of the of max{Xj : j = 0, . . . ,K(t)}, where K(t) is the number of independent and identically distributed observations (Xj) arriving up to the time t according to a general marked renewal cluster process. The results are illustrated by applying them in the context of some commonly used renewal cluster models such as the marked Hawkes process.