The talk will discuss some properties concerning algebraic numbers of small Weil height, more specifically properties (N) and (B) introduced by Bombieri and Zannier. A field of algebraic numbers has property (N) if it contains finitely many elements of bounded height, while it has property (B) if the height of its elements, when nonzero, is lower bounded by an absolute constant. While it is easy to see that number fields enjoy both properties, a generally difficult problem is to decide their validity for infinite extensions of the rationals. After surveying what is known in this area, I will present some results obtained in collaboration with Arno Fehm and some perpectives.