This text consists of two parts. In the first one we present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for

number fields and that « with moving targets » of Vojta) as an application of

Geometric Invariant Theory (GIT). Roth's Theorem is deduced from a general

formula comparing the height of a semi-stable point and the height of its

projection on the GIT quotient. In this setting, the role of the zero estimates

appearing in the classical proof is played by the geometric semi-tability of the

point to which we apply the formula.

In the second part we study heights on GIT quotients. We generalise

Burnol's construction of the height and refine diverse lower bounds of the

height of semi-stable points established to Bost, Zhang, Gasbarri and Chen.

The proof of Burnol's formula is based on a non-archimedean version of Kempf-Ness theory (in the framework of Berkovich analytic spaces) which completes

the former work of Burnol.