The theory of point processes constitutes an important part of modern stochastic process theory and is widely recognized as a useful and elegant tool for modelling. Point processes are well understood models and have applications in a wide range of applied probability areas, especially in risk theory which is important for understanding non-life insurance mathematics. It deals with the modelling of claims and gives answers on premium amount. Elegant mathematical analysis of the classical Cramér - Lundberg risk model has an important place in non-life insurance theory. The theory yields precise or approximate computations of the ruin probabilities, appropriate reserves, distribution of the total claim amount and other properties of an idealized insurance portfolio. In recent years, some special models have been proposed to account for the possibility of clustering of some events, for instance the Hawkes processes. We study asymptotic distribution of the total claim amount in the setting where Cramér - Lundberg risk model is augmented with a marked Poisson cluster structure. Marked Hawkes processes are then a special case and have an important role as the key example in our analysis. To make this more precise, we model arrival of claims in an insurance portfolio by a marked point process, say \[N = \sum_{k=1}^{\infty}\delta_{\tau_k,A^k}\] where \(\tau_k\)'s are non-negative random variables representing arrival times with some degree of clustering and \(A^k\)'s represent corresponding marks in a rather general metric space \(\mathbb{M}\). For each marked event, the claim size can be calculated using a measurable mapping of marks to non-negative real numbers, \(f: \mathbb{M}\) \rightarrow \mathbb{R}_{\geq 0}\) say. So that the total claim amount in the time interval \([0, t]\) can be calculated as \[S(t) = \sum_{\tau_k \leq t}f(A^k).\] We determine the effect of the clustering on the quantity \(S(t)\), as \(t\to \infty \), even in the case when the distribution of the individual claims does not satisfy assumptions of the classical central limit theorem. Besides new results regarding the case when second moments do not exist, we use different approach based on the limit theory for two dimensional random walks which stems from the classical Anscombe's theorem and not on martingale central limit theorem which was commonly used. We present the central limit theorem for the total claim amount \(S(t)\) in our setting under appropriate second moment conditions and prove a functional limit theorem concerning the sums of regularly varying non-negative random variables when subordinated to an independent renewal process. Based on this, we prove the limit theorem for the total claim amount \(S(t)\) in cases when individual claims have infinite variance. Moreover, we apply these results to three special models. In particular, we give a detailed analysis of the marked Hawkes processes which are extensively studied in recent years. In the last chapter we move our attention to the maximal claim size and present our results regarding limiting behaviour of maximum when claims belong to the maximum domain of attraction of one of the three extreme value distributions (Fréchet, Weibull and Gumbel). We also apply those results to three special models which we studied in previous chapter. Besides that, we try to clarify the notion of stochastic intensity which can be described in several different ways. The understanding of the stochastic intensity is important because of it's usage in the implicit definition of Hawkes processes.