The aim of this talk is to give an introduction to Teichmüller space. Roughly speaking, Teichmüller space is a space of geometric surfaces sharing the underlying topological surface. An important property is its relation to the so-called moduli space. This presentation has a connection to Anna Wienhard's colloquium talk. It will contain a review of concepts related to the topology of an orientable compact surface, such as its genus and fundamental group, and concepts related to its geometry, such as hyperbolic structures, conformal structures and the uniformization theorem. Teichmüller space parametrizes these structures. Further, it is related to representations of the fundamental group as a discrete group of isometries of the hyperbolic plane. As final outlook, we will discuss some results relating to Teichmüller spaces.