Let X be an algebraic curve of genus g⩾2 embedded in its Jacobian variety J. The Manin-Mumford conjecture (proved by Raynaud) asserts that X contains only finitely many points of finite order. When X is defined over a number field, Bogomolov conjectured a refinement of this statement, namely that except for those finitely many points of finite order, the Néron-Tate heights of the algebraic points of X admit a strictly positive lower bound. This conjecture has been proved by Ullmo, and an extension to all subvarieties of Abelian varieties has been proved by Zhang soon after. These proofs use an equidistribution theorem in Arakelov geometry due to Szpiro, Ullmo, and Zhang. Using more classical techniques of diophantine geometry, David and Philippon have given another proof which, moreover, provides an effective lower bound. In the talk, I will present the equidistribution statement, and his powerful generalization due to Yuan. I will then give the proof of the Bogomolov conjecture following Ullmo-Zhang. If time permits, I will also describe the proof of David and Philippon. I then plan to introduce the non-archimedean analogue of the equidistribution result and its application by Gubler to the Bogomolov conjecture over function fields.