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We discuss Mahler's work on Diophantine approximation and its applications toDiophantine equations, in particular Thue-Mahler equations, S-unit equationsand S-integral points on elliptic curves, and go into later developmentsconcerning the number of solutions to Thue-Mahler equations and effectivefiniteness results for Thue-Mahler equations. For the latter we need estimatesfor p-adic logarithmic forms, which may be viewed as an outgrowth of Mahler'swork on the p-adic Gel'fond-Schneider theorem. We also go briefly intodecomposable form equations, these are certain higher dimensionalgeneralizations of Thue-Mahler equations.

Mahler's work on Diophantine equations and subsequent developments