Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

I will describe results old and new regarding finiteness and infiniteness of integral points on arithmetic schemes, subject to analytic conditions. I will explain how the language of A-schemes and the use of theta-invariants, as developed in joint work with Bost, provides a geometric framework for this kind of questions, and how it relates to notions of affineness in Arakelov geometry.