This project gives a basic understanding of the difference between the Euclidean geometry and the non-Euclidean geometries (elliptic and hyperbolic). Furthermore the focus of the project is to describe the discovery of non-Euclidean geometry in the view of Kuhn’s paradigm shift as proposed in his book “The structure of scientific revolutions”(1962). This was done by matching the steps necessary for a paradigm shift described by Kuhn to the historical process of the discovery of non-Euclidean geometry. Analysing the steps, it was found that the paradigm shift could be split into four phases: a previous normal science, an anomaly, a crisis, and a new normal science. In this project the discovery of non-Euclidean geometry is explained as a revolution in the history of mathematics. Euclidean geometry can be seen as a previous normal science. The long discussion around the Fifth Postulate is an anomaly, followed by the crisis of attempting to expand geometry into including non-Euclidian theories during the 19th century with Gauss, Lobatchewsky and Bolyai. The culmination finally came with Riemann, Beltrani and Klein in a new paradigm with a synthetic view, where different assumptions about the fifth postulate result in different geometries.