  This is an introduction to the topics of the title, from the 2017 Grenoble
Summer school on Arakelov geometry and arithmetic applications. We review
Arithmetic intersection numbers, explain the definition of the height of a
variety and its properties, both in the framework of classical Arakelov
geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness
and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan,
both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof
of the Bogomolov conjecture over number fields.
