  We review the basic ideas and some theoretical models behind the concept of
fullerene-like structures made of $\alpha$-particles. The possibility of such a
peculiar nuclear shape developing in a double magic superheavy nucleus with
$Z$=120 was mentioned for the first time in the literature by Walter Greiner
and then constantly defended by him. We provide estimates of the energy of such
metastable states within the liquid-drop model. In the second part of this
paper we discuss a simple model rooted in the nontopological soliton model
consisting of a complex scalar field describing a finite system of $\alpha$
bosons with non-linear self-interactions coupled to the electromagnetic field.
We demonstrate that this model predicts density depletion in the central region
of a soliton-like structure for a range of model parameters.
