Aspects of the Bogomolov conjectures

The original Bogomolov conjecture (proved by Ullmo) asserts that a projective curve of genus greater than 1 over a number field has only finitely many points of small Neron-Tate heights. We will introduce this conjecture and the recent developments on other versions of it, including the geometric Bogomolov conjecture and the uniform Bogomolov conjecture.