In this study we observed the Cramér-Lundberg model in the theory of risk. It consists of the assumption that the aggregate claims process is a compound Poisson process. In the first chapter we defined a compound Poisson process, and gave relevant definitions and extracts that we would later use. We defined a surplus process as opposed to the sum of the insurer’s surplus at time 0 and the insurer’s rate of premium income per unit time and aggregate claims at time t. We deduced a formula for the probability that the surplus process is less than 0 at some time t, and gave an upper bound for that probability. In addition to the ruin probability, we also defined the survival probability. Since in some cases the formula for the probability of ruin can not be explicitly calculated, we established some approximations thereof. In the third chapter, we define two approximations, the De Vylder approximation and the 4MGDV approximation. In case of exponentially distributed individual claim amounts, we observed that the De Vylder approximation gives exact results. We also concluded that the 4MGDV approximation gives better results than the De Vylder approximations.