Diophantine geometry and analytic spaces

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.