The relationship between the structure and the compressive strength of carbon fibres has been studied in detail. In order to determine the compressive strength, a combination of single-fibre composite tests and Raman spectroscopy was employed. It was found that the compressive stress-strain curves showed nonlinear behaviour, with modulus softening in compression. The compressive strengths for the fibres with a modulus a parts per thousand yen400 GPa were measured as a parts per thousand currency sign2 GPa and those with a modulus < 400 GPa were > 2 GPa. We have introduced a model to explain this behaviour that assumes that the fibres behave as composites consisting of both crystallites and amorphous carbon. It is suggested that the compressive strength is controlled by the critical stress for kinking the crystallites in the fibres. Hence, the compressive strength of carbon fibres is found to depend upon the shear modulus of the fibres and the orientation of the crystallites within them.