Many applications of carbon nanotubes require the exfoliation of the nanotubes to give individual tubes in the liquid phase. This requires the dispersion, exfoliation and stabilisation of nanotubes in a variety of liquids. In this paper we review recent work in this area, focusing on results from the author's group. We begin by reviewing stabilisation mechanisms before exploring research into the exfoliation of nanotubes in solvents, by using surfactants or biomolecules and by covalent attachment of molecules.The concentration dependence of the degree of exfoliation in each case will be highlighted. In addition we will discuss research into the dispersion mechanism for each dispersant type. Most importantly we have compared dispersion quality metrics for all dispersants. From this analysis, we conclude that functionalised nanotubes can be exfoliated to the greatest degree. Finally, we review the extension of this work to the liquid phase exfoliation of graphite to give graphene.