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We review the basic ideas and some theoretical models behind the concept offullerene-like structures made of $\alpha$-particles. The possibility of such apeculiar nuclear shape developing in a double magic superheavy nucleus with$Z$=120 was mentioned for the first time in the literature by Walter Greinerand then constantly defended by him. We provide estimates of the energy of suchmetastable states within the liquid-drop model. In the second part of thispaper we discuss a simple model rooted in the nontopological soliton modelconsisting 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 regionof a soliton-like structure for a range of model parameters.

The Fullerene-like Structure of Superheavy Element Z=120 (Greinerium)-a tribute to Walter Greiner