The behaviour of fullerenes C{sub 60}, C{sub 70} and C{sub 60}I{sub 4} under pressure has been studied. Their electrical resistivity follows a 3-D variable range hopping behaviour at low pressures. A transition to a less conducting state has been observed at pressures higher than 15 GPa. In the case of C{sub 60} or C{sub 60}-C{sub 70} mixtures attempts of stabilizing the high pressure phase have resulted in the collapse of the fullerene structure to polycrystalline diamond or amorphous carbon, depending on the pressure and on the conditions of hydrostaticity. This is the first time that any carbon phase transforms to diamond at room temperature, suggesting a potential application to industrial diamond synthesis. The nature of the obtained amorphous carbon phase, possibly sp{sup 3} in nature, is discussed.