The failure of fibre composites under high rates of test was studied in detail, using a fracture mechanics approach to determine values of the fracture energies. The present paper, Part III of the series, considers the fracture of two different fibre composite materials under mixed-mode tensile and shear (i.e. mixed-mode I/II) and shear (i.e. mode II) loadings. Part I considered the experimental aspects of the mode I fracture of the fibre composite materials, and of adhesive joints. Part II analysed the dynamic effects which are invariably associated with high-rate tests, and showed how these effects influence the observed behaviour of the test specimens.