The label carbon nanostructure magnetism joins two notions that do not seem to go together easily, carbon and magnetism. The main carbon allotropes, after all, are known to be non-magnetic. This appears to be true not only about the wellknown solid phases diamond and graphite, but also with respect to the nanoscopic phases manufactured first in the eighties of the last century, and later: fullerenes and carbon nanotubes, single- and few-layer graphene. While intrinsic magnetism is the rule in the d- and f-blocks of the periodic table, it is extremely unusual in the second period, containing light elements with p electrons in their valence shells. Magnetic derivatives from carbon-based nanostructures, such as metallofullerenes with finite magnetic moments in their ground state have been known for decades, but the magnetism of these composites is inherited from elements different from carbon, such as lanthanide atoms with high spins localized in their 4f shells.

By the beginning of this century, however, sightings of intrinsic magnetism in carbon complexes became increasingly frequent and made headlines, not rarely heralded with adjectives like surprising [1], unexpected [2] or exotic [3]. While in the meantime, the initial surprise about carbon magnetism has somewhat worn off, astonishing discoveries continue to be made in this field, such as the first experimental demonstration of spin transport in graphene at the micrometer scale [4] or the first detection of strong spin-orbit coupling in carbon nanotubes [5].

On the other hand, magnetism in carbon nanostructures also continues to be a topic in tension. Foremost, the proposal of intrinsic carbon magnetism due to net magnetic moments at edges or vacant sites in carbon networks is charged with controvery. Proponents point not only at a large body of theoretical and computational work, predicting these effects (for an overview, see [6]), but also at numerous experiments that appear to confirm these predictions. Detractors call attention to the great difficulty of reliably separating signatures of intrinsic magnetism from artifacts due to small admixtures of magnetic impurities [3] and refer to the rather marginal, if not vanishing, net effects yielded by some recent experimental examinations of carbon magnetism [7].

In view of the ongoing debate, it would be premature to state that magnetism in carbon nanostructures is a firmly established discipline within condensed-matter physics or nanoscience.