A so-called translation surface is a surface obtained from a euclidean polygon with pairwise parallel sides by gluing the sides with euclidean translations. Alternatively, it is a Riemann surface together with a holomorphic one-form. I will give an elementary geometric overview of some recent results of Eskin, Filip, Mirkzakhani and explain how this together with some classical differential geomtric tools can be used to dreive some algebraic-geometric properties of the moduli space of translation surfaces (finiteness of algebraically primitive Teichmüller curves) which in turn can be translated into properties of the mapping class group.