The paper addresses a compressive-failure theory for polymer-matrix nanocomposites in the case where failure onset is due to microbuckling. Two approaches based on the three-dimensional linearized theory of stability of deformable bodies are applied to laminated and fibrous nanocomposites. According to the first approach (continuum compressive-failure theory), nanocomposites are modeled by a homogeneous anisotropic medium with effective constants, including microstructural parameters. The second approach uses the piecewise-homogeneous model, three-dimensional relations for fibers (CNT) and matrix, and continuity conditions at the fiber-matrix interface. The compressive-failure theory is used to solve specific problems for laminated and fibrous nanocomposites. Some approximate failure theories based on the one-and two-dimensional applied theories of stability of rods, plates, and shells are analyzed Keywords: nanocomposites, CNT fiber, polymer matrix, compressive failure, microbuckling, three-dimensional linearized theory of stability of deformable bodiesIntroduction. Today's literature on the fracture mechanics of composites considers the paper [17] to be the first to describe, in 1960, fiber microbuckling as a compressive failure mechanism for unidirectional fibrous composites. In the years that followed, several authors set forth different approximate models for the quantitative and qualitative description of this mechanism. These models are based on a number of assumptions and hypotheses, of which the following are worth mentioning: no (neglected) subcritical stresses in the matrix, use of applied one-and two-dimensional theories of stability of rods and plates to study microbuckling, modeling of the matrix by a one-dimensional elastic object, etc.The paper [14] was apparently the first to propose, in 1965, a highly approximate model for the quantitative description of microbuckling in composites within the framework of a plane problem (in fact, the fibrous composite was modeled by a laminated composite), using the above-mentioned assumptions and hypotheses (the same results were reported in [14]). Despite the highly approximate model, the results from [14] were used in many publications, including the seven-volume collective monograph [13], and are generally recognized and widely cited. In the literature on the fracture mechanics of composites (see, e.g., [15]), these results are named the Dow-Gruntfest-Rosen-Schuerch theory, after the authors of the first publications [14,17,45,46].Thus, to describe the failure mechanism in question [17], we need a stability theory for unidirectional fibrous composites (Fig. 1) or laminated composites (Fig. 2) subjected to axial compression. The paper [17] addressed a fibrous composite; thus, it is expedient to construct a stability theory for the material represented in Fig. 1. In this connection, a stability theory for laminated composites (Fig. 2) is to be developed to attain two ends (two different cases). In the first case, a stability theory is needed to describe the f...