This is a review of the results of several selected theoretical studies concerned with the localized kinking, or microbuckling, of aligned fiber composite materials subjected to compression in the fiber direction. Compressive kinking is of primary concern in polymer matrix composites, for which kinking failure can limit the compressive strength to a value that is usually much lower than the tensile strength. A similar situation can occur in carbon matrix fiber composites. Compressive kinking failure may be understood on the basis of an elementary theoretical approach that ignores the influence of bending resistance of the reinforcing fibers, but takes into account the nonlinearity of composite constitutive relations as well as the effects of initial imperfections in fiber alignment. Kink bands bounded by fiber breaks are produced by deformations that occur after the attainment of peak compressive loads. The theoretical calculation of the widths of such kink bands does require consideration of fiber bending resistance; on the other hand, the results for kink width are not sensitive to the sizes of initial fiber misalignments. Progress in the study of the following additional kinking topics is summarized briefly: correlation of static kinking strength and random fiber misalignment; effects of shear and transverse loads on static kinking; viscoelastic and creep kinking; kinking fatigue; and a phenomenological theory of kinking “toughness”.