This thesis presents the basic concepts of risk theory that models two sources of non-life insurance uncertainty: how many claims an insured person will report and what are the amounts of those claims. Accordingly, claim number process and total claim amount process are defined. Together, these two processes form a constitutive part of the risk process that monitors the relation between premiums and claims paid. The paper is divided into five chapters. The first two chapters describe the basic properties of the mathematical models for the claim number process - homogeneous Poisson process and renewal process. The third chapter provides an overview of the claim size distributions with particular emphasis on heavy-tailed distributions - regularly varying and subexponential distributions. The fourth chapter explains the basic model properties for the total claim amount process - Cramér-Lundberg model and renewal model. Part of the chapter is also devoted to deter- mining the distribution of the total claim amount up to a certain moment \(t\). The last chapter presents the fundamental results of the ruin theory whose main focus is the risk process.