The discovery in 2005 of superconductivity in YbC6 and CaC6, with substantially higher critical temperatures than the previously observed among the family of the graphite intercalation compounds, has largely renewed the interest for these well known lamellar compounds. Indeed, these critical temperatures reach 6.5 and 11.5 K respectively for ytterbium- and calcium-graphite phases. It was consequently interesting to collect all the informations concerning the superconductivity of these compounds from the discovery of this phenomenon observed in the heavy alkali metals graphite intercalation compounds in 1965, insisting particularly on the recent advances in this research field. After a general introduction that describes all the carbon materials, which are extremely various with dimensionalities varying from 3 to 0, leading to their large aptitude for the insertion/intercalation reactions, the authors widely developed the case of graphite: chemical bonds, crystal and electronic structures, anisotropy and ability to become a host structure. The authors insist on its strong anisotropy of chemical reactivity that allows the synthesis of very numerous intercalation compounds. The distinctive features of the intercalation reaction into graphite are reviewed (systematic charge transfer, staging, etc...) and are particularly developed in the case of the donor-type intercalation compounds, among which is precisely observed the superconductivity. For the latter, the various synthesis methods are successively described, showing the best route to use in order to obtain each type of compound. Then the authors review with detail the binary compounds, emphasising their distinctive crystal and electronic structures and also their transport properties. The authors describe the superconductivity of all the compounds belonging to this family and show this property. In the last part, the authors compare these superconducting binary intercalated graphite compounds with other lamellar superconductor: magnesium diboride. The ternary compounds are then studied, and the poly-layered nature of their intercalated sheets is given special attention. Their distinctive electronic structure is presented and their superconducting properties are described.