Carbon centres in typical organic molecules have a coordination number that can reach a maximum of four, in which case the bonded atoms are situated at the vertices of a tetrahedron. Exceptions to those two structural rules have been posited and examined for decades, and planar tetracoordinate carbon (ptC) species are notable molecules that violate the second rule. There is continued interest in experimental and theoretical studies of ptCs, as well as emerging molecules that contain planar pentacoordinate carbon (ppC) and planar hexacoordinate carbon (phC) atoms, species that violate both structural rules. This Review describes recent progress in the theoretical prediction of viable entities that contain ppC centres. The first such molecule reported, the D5h-symmetric ppC species CAl5+, was followed by a series of predicted ppC species that could be obtained by substituting the Al centres for other heteroatoms. More complicated ppC systems have also been suggested, including metallocene-stabilized ppCs and quasi-ppCs embedded within cage structures or 2D materials. To date, computational studies have identified at least 65 local and 39 global minimum energy structures that contain ppCs or quasi-ppCs. The general design principles for ptC-centred candidate structures include delocalization of the central C 2pz lone electron pair, ensuring an 18 valence electron count and allowing for strong electron delocalization. These principles have been extended to ppC systems with some success. It is hard to predict the extent to which the coordination number of planar C can be increased because it depends not only on the valence and size of C but also on the size of the atoms bonded to it and the mode of bonding. Although a few energetically low-lying planar hexacoordinate and heptacoordinate C species have been identified computationally, none have been observed experimentally.