A bundle of carbon fibers has a value A obtained from a nonlinear approximation formula of a stress σ-strain ε curve in a tensile strength test of resin-impregnated strands and an orientation parameter Π (%) of crystallites in a wide-angle x-ray diffraction measurement which satisfy a predetermined relational expression, and has tensile strength with a predetermined value or more, and tensile modulus within a predetermined range and a product E×d/W of a ratio d/W of a single-fiber diameter d to a loop width W just before loop fracture evaluated by a single-fiber loop test and a tensile modulus E of the strands has a predetermined value or more, or apparent single-fiber stress has a predetermined value or more when the number of fiber breaks by a single-fiber fragmentation method for a single-fiber composite is 0.30 breaks/mm and when the number of the fiber breaks by the single-fiber fragmentation method for the single-fiber composite is 0.30 breaks/mm, the number of fiber breaks by a double-fiber fragmentation method for the single-fiber composite is within a predetermined range.