It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is 


What can be said about a finite group $G$ for which $\chi(x)=0$ for all irreducible character $\chi$ and elements $x\in G\setminus Z(\chi)$?


Here, $Z(\chi)=\{g\in G:|\chi(g)|=\chi(1)\}$. So, $Z(\chi)/\ker\chi=Z(G/\ker\chi)$.



Groups with many vanishing elements
These appear to be the groups that I was asking about in [this older MO question](https://mathoverflow.net/questions/56380/characters-on-a-finite-group-with-extremal-behaviour) -- see in particular the answers by Isaacs and Ladisch.


