The formation of both positive and negative fullerene ions from C-30(-) to >C-650(+) in premixed low-pressure flames of acetylene, 1,3-butadiene, benzene, and naphthalene with oxygen is reported and reviewed. For benzene, the formation both of uncharged fullerenes and of their ions is compared. Fullerenes are formed in the soot-free oxidation zone and also at the beginning of soot formation in the burned gas, Their increase is strongly correlated with the concentration of polycyclic aromatic hydrocarbons (PAHs) in the flame and ceases when the PAHs are consumed or oxidized. Without the presence of PAHs soot is not a source of fullerenes. Species which already have the necessary number of C atoms but still contain a few H atoms could be detected as fullerene precursors. They were most sensitively detected in the negatively charged state. When the temperature decreases in the burned gas, fullerene ions add H atoms. The extent of acquisition of hydrogen depends on the occurrence of adjacent pentagons, which is always the case in fullerenes <C-60. A mechanism of formation is proposed beginning with the reaction of two PAH molecules of sufficient size, but which do not necessarily contain pentagons. In contrast to normal PAH growth in flames, most of the hydrogen is split off from the periphery of the two PAHs by a kind of concerted zipper mechanism, through which simultaneously the necessary number of pentagons are generated. During this process the latter are shifted into energetically more favourable positions. A generation by repetitive bimolecular reactions of PAHs with acetylene or another small aliphatic molecule can be excluded.