Abstract3D failure criteria for laminated fibre-reinforced composites, based on a physical model for each failure mode and considering nonlinear matrix shear behaviour, are developed. Special emphasis is given to compression failure. The physical model for matrix compression failure is based on the Mohr-Coulomb criterion and also predicts the fracture angle. For fibre kinking, an initial fibre-misalignment angle is considered to trigger failure, due to further rotation during the compressive loading. The plane where the kinking takes place is predicted by the model, as well as the kink-band angle. Applications are presented that validate the model against experimental data.