The most thermodynamically stable form of carbon under normal conditions is graphite, which is a stack of loosely bound graphene sheets, flat atomic honeycomb lattices. Each atom is linked to three neighbours, the fourth valence electron forming the -system, i.e. the carbon atoms are in the sp²-hybrid state. Geometric defects in graphene sheets can lead to a closed 3D structure. The role of such defects is played by the pentagonal faces (five-membered rings), the most common entities in the cyclic organic compounds along with the hexagonal cycles. The Euler theorem dictates that any polyhedron with three edges (bonds) to each vertex must contain 12 pentagons, regardless of the number of hexagonal faces. Thus, the smallest possible fullerene is the dodecahedron C₂₀.

However, the high curvature of C₂₀ and other smaller structure carbon atoms is too unsuitable for sp²-hybrid state of carbon, which prefers flat coordination. Therefore, the smallest stable pristine fullerene is C₆₀, a truncated icosahedron where all pentagonal faces are separated from each other by hexagonal ones.

This fact is a manifestation of the isolated pentagon rule (IPR), well-known in fullerene chemistry, obeyed by all pristine fullerenes available in reasonable quantities. Because of the IPR, C₆₀ fullerene is followed by C₇₀ since there are no intermediate structures with isolated pentagons. Beyond C₇₀, there exist fullerenes with any even number of carbon atoms, so-called higher fullerenes/ Furthermore, starting with C₇₈, several stable isomers are observed for each carbon skeleton.

Fullerene synthesis is predominantly carried out via the arc-discharge technique, but also by means of electron-beam or laser ablation of graphite in the helium atmosphere. The resulting soot that condenses on the cold surface of the reactor is collected and extracted with boiling toluene, benzene, xylene or other aromatic solvents. The black residue after evaporation contains larger amounts of C₆₀ and C₇₀ and smaller admixtures of higher fullerenes. Depending on the synthetic parameters, the ratio between C₆₀ and C₇₀ can vary, but usually the content of C₆₀ is several times higher than that of C₇₀. Among the higher fullerenes, C₈₄, C₇₆ and C₇₈ prevail. In general the abundance decreases with the molecular size due to the decrease in probability of the assembly of larger structures from the initial small carbon building blocks.

C₆₀ is the most readily available and broadly studied fullerene. Its molecule, where all the atoms are equivalent due to its high symmetry, has a spherical shape with a distance from the centre to the atomic nuclei of about 0.36 nm and a van der Waals radius of about 0.5 nm. C₆₀ forms molecular crystals with the molecules being localisised in the nodes of a face-centred cubic, i.e. three-layered close-packed, lattice (see fullerite). At high temperatures, C₆₀ sublimates without forming the liquid phase. C₆₀ is most soluble in aromatic substances and solvents such as carbon disulphide, whereas it poorly dissolves in polar solvents. Physical and chemical properties of C₇₀, which has an elongated ellipsoidal shape, and of higher fullerenes are similar to those of C₆₀.

In terms of chemistry, fullerenes provide broad opportunities for various kinds of derivatisation. Encapsulation of atoms and small clusters inside the carbon cage results in endohedral fullerenes, with metallofullerenes being the most interesting of them (e.g., La@C₈₂, Sc₃N@C₈₀). Substitution of the cage carbon atoms result in heterofullerenes (e.g., C₅₉B, C₄₈N₁₂, C_59-2nFe, where n = 0-10, or C_60,70M_x, where M = Rh, Ir and x = 3-15 for Rh and x = 2 - 5 for Ir). The products of exohedral addition constitute the largest family, since virtually all the carbon atoms of the fullerene cage is a readily available reaction site. In particular, there exist products of fullerene hydrogenation and halogenation, as well as of their functionalisisation with organic radicals and rings/ Also, fullerene-containing polymers and polyspheroid fullerene derivatives have been produced. Chemical functionalisisation commonly yields mixtures with a broadly varying number of addends and a complex isomeric composition. For example, in the case of C₆₀ one can attach up to 48 substituents without destroying the carbon skeleton (e.g., C₆₀F₄₈).

Fullerenes are the focus of practical interest in many areas. In the condensed phase, fullerenes and their derivatives can be regarded as n-type semiconductors (with a bandgap of about 1.5 eV for C₆₀). They readily absorb radiation in the ultraviolet and visible region. The spherically conjugated -systems of fullerenes give rise to their high electron-withdrawing ability (the electron affinity of C₆₀ is 2.7 eV, in many higher fullerenes it exceeds 3 eV and may be even higher in some derivatives). This makes fullerenes promising compounds for photovoltaics; actively developed are fullerene-based donor-acceptor dyads for application in solar cells (there are samples with 5.5% efficiency), photosensors, and other molecular electronic devices. Biomedical fullerene applications constitute yet another promising research direction, in particular, development of antimicrobial and antiviral agents, agents for photodynamic therapy, etc.