Among all elements, C is the basis of entire life. The whole branch of chemistry - the organic chemistry - is devoted to the study of C-C bonds and different molecules originating from them. Carbon is only 4-valent element able to produce long homoatomic stable chains or different 4-regular nets. The other 4-valent candidate for this could be only Si, with its reach chemistry beginning to develop. After diamond and graphite - the hexagonal plane hollow shell, in 1985 was first synthesized by H.W.Kroto, R.F.Curl and R.E.Smalley the spherical closed pentagonal/hexagonal homoatomic shell: the fullerene C60. Except from this, it possesses some another remarkable properties: the rotational symmetry of order 5, from the geometrical reasons (according to Barlow "crystallographic restriction theorem") forbidden in crystallographic space or plane symmetry groups, and highest possible icosahedral point-group symmetry. After C60, different fullerenes (e.g. C70. C76. C78. C82. C84 etc.) are synthesized, opening also a new field for research of different potentially possible fullerene structures from the geometry, graph theory or topology point of view.