This course gives an introduction to the theory of Diophantine Approximations with a special emphasis on its impact to the eld of Diophantine Equations. It relies mainly on the nicely written books [9], [29] as well as the more general introduction [8] and the classics [21], [35]; historical details can be found in [11], [48] and on www:history:mcs:st− andrews:ac:uk. Finally, we refer to [4] for modern perspectives in this amazing and growing eld.