  We discuss Mahler's work on Diophantine approximation and its applications to
Diophantine equations, in particular Thue-Mahler equations, S-unit equations
and S-integral points on elliptic curves, and go into later developments
concerning the number of solutions to Thue-Mahler equations and effective
finiteness results for Thue-Mahler equations. For the latter we need estimates
for p-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's
work on the p-adic Gel'fond-Schneider theorem. We also go briefly into
decomposable form equations, these are certain higher dimensional
generalizations of Thue-Mahler equations.
