  We first recall the connection, going back to A. Thue, between rational
approximation to algebraic numbers and integer solutions of some Diophantine
equations. Next we recall the equivalence between several finiteness results on
various Diophantine equations. We also give many equivalent statements of
Mahler's generalization of the fundamental theorem of Thue. In particular, we
show that the theorem of Thue--Mahler for degree $3$ implies the theorem of
Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of
Siegel on the rational integral points of the projective line $\P^1(K)$ minus
$3$ points. Finally we extend our study to higher dimensional spaces in
connection with Schmidt's Subspace Theorem.
