The aim of this thesis was to describe models for claim sizes in insurance and appropriate probability distributions, with heavy tail distributions playing the largest role. The first chapter talks about the concept of reinsurance, what it is and what are the situations where it is necessary. Also, the basic forms of proportional and disproportionate reinsurance are listed, as well as their advantages and disadvantages. The sequel of the thesis discusses different approaches on what we might righteously call a large claim and how one can perhaps distinguish it from others, so in this context we define distribution classes such as regularly varying, subexponential and long tail distributions with their basic properties and examples. The so-called Fisher-Tippett-Gnedenko theorem, the basic result of the extreme-value theory is introduced, which characterizes the limit distribution for maximum of random variables. For all three cases the maximum domains of attraction are derived and the generalized Pareto distribution that plays a major role in the theory of extreme values is defined. Finally, examples of the most important distributions used in modelling reinsurance claims with an emphasis on one heavy tail are given (Pareto, lognormal, Weibull(α < 1)...), and their distribution and density functions are visually shown compared to other distributions