On the fundamental theorem of compact and noncompact surfaces

In this article an overview of the history of surface topology is given. From the Euler-formula, to the Kerekjarto Theorem we follow the development process of the fundamental theorem of compact and non-compact surfaces. We refer to the works of Riemann, Mobius, Jordan, Klein and others, but our main focus point is to show the work of the Hungarian mathematician, Bela Kerekjarto.