The high pressure experiments done on fullerenes are reviewed. C 60 has found to be stable up to about 20 GPa at room temperature and hydrostatic conditions. Application of stronger, or non-hydrostatic, pressures at room temperature can induce the formation of a partially sp3 bonded phase, that apparently conserves the fullerene cage. Extreme non-hydrostatic compressions above about 15 GPa can, though, break down the cage and produce amorphous or cubic diamond. Destruction of the cage at high temperatures has also been observed, but the resulting product is amorphous sp2 material. A preliminary pressure-temperature phase diagram for C 60 is proposed.