  Synthesis of new elements at the upper border of the charts of nuclei and
investigation of their decay properties and nuclear structure has been one of
the main research topics in low energy nuclear physics since more than five
decades. Main items are the quest for the heaviest nuclei that can exist and
the verification of the theoretical predicted spherical proton and neutron
shells at Z = 114, 120 or 126 and N = 172 or 184. The scope of the present
paper is to illustrate some technical and physical aspects in investigation of
the heaviest nuclei ('superheavy nuclei') and to critical discuss some selected
results, which from a strict scientific point of view are not completely clear
so far, making partly also suggestions for alternatively interpretations. A
complete review of the whole field of superheavy element research, however, is
out of the scope of this paper.
