It is natural to ask for stronger versions of Liouville’s theorem. Only a slight improvement would imply the ﬁniteness of integral solutions of certain important diophantine equations, so-called Thue equations. First improvements of Liouville’s theorem were made by Thue, Siegel, and Dyson. The most far-reaching extension was found by Roth for which he was awarded a Fields medal at the 1958 International Congress of Mathematicians at Edinburgh. In this chapter we will give a proof of this deep and far-reaching highlight in the theory of diophantine approximations.