  This text is the write-up of a talk at the Bellairs Workshop in Number Theory
on Tropical and Non-Archimedean Geometry that took place at the Bellairs
Research Institute, Barbados, in May 2011. The goal of this text is to present
recent work by in Diophantine Geometry over function fields due to Gubler and
Yamaki, where analytic geometry in the sense of Berkovich plays a significant
place. I also give an introduction to basic concepts and notions on Diophantine
Geometry, such as heights, the Manin-Mumford conjecture, the Bogomolov
conjecture, and its proof by Ullmo and Zhang.
