Most insurance companies deal with reinsurance. One of the problems they have to solve is: What reinsurance treaty is optimal for their company? Optimal reinsurance for large portfolios has, during the past decades, been given much more attention in the academic world than individual claims have. In this thesis we will investigate the optimal reinsurance contract for the individual claims case. The thesis will start by introducing some basic concepts in reinsurance. There will be a brief explanation of reinsurance mathematics. We will establish mathematical formulations of the different types of reinsurance contracts and the optimality criteria. Then we will see some of the existing literature on the topic and show some own results. This is going to point us toward the optimal reinsurance contract: the non-proportional a x b contract with retention limits a and b where b is infinite. We will introduce the reader to the Panjer recursion. The recursion will be used as a numerical tool to simulate the a x b contract. We will vary different key parameters and see how they affect the criteria and the retention limits. These results will back our assumption of the a x b contract with infinite b as the optimal one.