Abstract: Given an algebraic variety over the rational numbers, what can we say about the set of its rational points? In this talk we discuss questions around the existence of rational points, obstructions to the existence of rational points and the quantitative distribution of rational points on Fano varieties. In the second part of the talk we consider the more general question of studying rational points near manifolds, we discuss connections with Diophantine geometry and applications to Diophantine approximation.