abstract: In connection with the Lehmer problem, the notion of fields with the so called Bogomolov property has emerged (E. Bombieri, U. Zannier). These are fields containing no points of small height excluding torsion points (for the multiplicative group) and zero. We shall give a description of some the present pictures describing some recent progresses (joint with F. Amoroso and U. Zannier) and suggesting a few open questions.