Currently, polymer composite materials are widely used in the designs of advanced aircrafts, which significantly lightens the construction weight while maintaining its strength and stiffness characteristics. Despite the fact that there is a fairly large number of works on the study of such structures strength, the issues of their strength and stability under conditions of the initial nonlinear stress-strain state remain unsolved. The latter is especially necessary for aircraft fuselage structures, in which the composite skin stability loss is unacceptable. Methods of calculation for strength and stability of composite structures taking into account the nonlinearity of the initial stress-strain state are currently underdeveloped. Therefore, the development of reliable and efficient methods for analysis of shells made of composite materials is undoubtedly an urgent task. The most suitable method in this case is the finite element method. Its advantages are its versatility, physicality and unlimited applicability to complex structures under arbitrary loading. The application of the finite element method to the analysis of shells is associated with significant difficulties due to the thickness and curvature of the shell. The building of effective finite elements of shells is also an urgent problem to this day. Most of the developed finite elements are elements of circular cylindrical, conical or spherical shells. In the present work, the problem of strength and stability of cylindrical composite shells under arbitrary loading is solved by finite element methods and Newton-Kantorovich linearization. Finite elements of non-circular cylindrical composite shells and reinforcement elements of natural curvature, developed by the authors on the basis of the Timoshenko hypothesis, are used, in the approximation of displacements of which their rigid displacements (displacements of finite elements as a rigid body) are explicitly identified. Critical loads are determined in the process of a geometrically nonlinear problem solving using the matrix triangulation method and Sylvester’s criterion. The shapes of the shells in the prebuckling state and their buckling shapes are also calculated. The stability of a circular cylindrical shell made of a polymer composite material has been investigated under various types of loading: torsional and bending moments, edge compressive and transverse forces, and external pressure. The effect of the monolayers lay-up angles, deformation non-linearity on the critical loads of the shell instability has been determined.