Appears in collection : Journée sous-riemannienne 2017

The Besicovitch covering property originates from works of Besicovitch about differentiation of measures in Euclidean spaces. It can more generally be used as a usefull tool to deduce global properties of a metric space from local ones. We will discuss in this talk the validity or non validity of the Besicovitch covering property on stratified groups equipped with sub Riemannian distances (Carnot groups) and more generally on graded groups equipped with homogeneous distances. We will illustrate these results with explicit examples in the Heisenberg group. We will also discuss some consequences related to the theory of differentiation of measures on sub-Riemannian manifolds. Based on joint works with E. Le Donne and S. Nicolussi Golo.