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This tag should be applied to questions about the various allotropes of carbon and their physical or chemical properties. Most important modifications are diamond, graphite, fullerenes, graphene and nanotubes.

One of the main courses of chemistry research in the current decade is devoted to exploring the different allotropes of carbon. Carbon can form a variety of intramolecular bonds with itself, which allows its allotropes to have very special properties.

Most important allotropes are

• Diamond
In the bulk material, every carbon atom is covalently bonded to four other carbon atoms, giving it a tetrahedral coordination sphere. It is a very hard material and therefore has many applications in industry and also as jewelry.
From the IUPAC goldbook:

An allotropic form of the element carbon with cubic structure (space group $O_{h}^{7} − F d 3 m$) which is thermodynamically stable at pressures above $6~\mathrm{GPa}$ at room temperature and metastable at atmospheric pressure. At low pressures, diamond converts rapidly to graphite at temperatures above $1900~\mathrm{K}$ in an inert atmosphere. The chemical bonding between the carbon atoms is covalent with $\ce{sp^3}$ hybridization.
Note:
There is also a hexagonal diamond-like structure of the element carbon (lonsdaleite).

• Graphite
Graphite is the most common allotrope of carbon and its thermodynamically stable form at standard temperature and pressure. It is also a good electrical conductor.
From the IUPAC goldbook:

An allotropic form of the element carbon consisting of layers of hexagonally arranged carbon atoms in a planar condensed ring system graphene layers The layers are stacked parallel to each other in a three-dimensional crystalline long-range order. There are two allotropic forms with different stacking arrangements, hexagonal and rhombohedral. The chemical bonds within the layers are covalent with $\ce{sp^2}$ hybridization and with a $\ce{C–C}$ distance of $141.7~\mathrm{pm}$. The weak bonds between the layers are metallic with a strength comparable to van der Waals bonding only.
Note:
The term graphite is also used often but incorrectly to describe graphite materials, i.e. materials consisting of graphitic carbon made from carbon materials by processing to temperatures greater than $2500~\mathrm{K}$, even though no perfect graphite structure is present.

• Graphene layer
A special form of graphite. According to the IUPAC goldbook:

A single carbon layer of the graphite structure, describing its nature by analogy to a polycyclic aromatic hydrocarbon of quasi-infinite size.
Note:
Previously, descriptions such as graphite layers, carbon layers or carbon sheets have been used for the term graphene. Because graphite designates that modification of the chemical element carbon, in which planar sheets of carbon atoms, each atom bound to three neighbours in a honeycomb-like structure, are stacked in a three-dimensional regular order, it is not correct to use for a single layer a term which includes the term graphite, which would imply a three-dimensional structure. The term graphene should be used only when the reactions, structural relations or other properties of individual layers are discussed.

• Fullerene
Also known as buckyballs. All fullerenes are molecules composed entirely of carbon and forming sphere- or cage-like structures of varying size. According to the IUPAC goldbook:

Compounds composed solely of an even number of carbon atoms, which form a cage-like fused-ring polycyclic system with twelve five-membered rings and the rest six-membered rings. The archetypal example is [60]fullerene, where the atoms and bonds delineate a truncated icosahedron. The term has been broadened to include any closed cage structure consisting entirely of three-coordinate carbon atoms.

• see Wikipedia for a more complete list of carbon allotropes

This tag should be applied to questions about the any of the allotropes of carbon and their physical or chemical properties.