Let G be a finite group, and let \(\textrm{Irr}(G)\) denote the set of the irreducible complex characters of G. An element \(g\in G\) is called a vanishing element of G if there exists \(\chi \in \textrm{Irr}(G)\) such that \(\chi (g)=0\) (i.e., g is a zero of \(\chi \) ) and, in this case, the conjugacy class \(g^G\) of g in G is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group G such that every non-linear \(\chi \in \textrm{Irr}(G)\) vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.
