A folding tonneau cover apparatus including a cover assembly and a support frame assembly. The cover assembly having a plurality of rigid panels interconnected in series by a series of flexible hinges preferably made from a laminated woven fabric material. The rigid panels are preferably stiffened by elongated support members secured to the undersides of the respective panels and the top surfaces of portions of the rigid panels are preferably downwardly concave. The cover assembly is secured to the support frame assembly to cover a cargo box of a pickup truck by a plurality of locking members when the support frame assembly is secured to sidewalls of the cargo box. Methods of making a tonneau cover apparatus and methods of folding and unfolding the cover assembly are also disclosed.