The current status of research on the mechanical properties of carbon nanotubes (CNTs) and composites containing them is reviewed. The structural features of CNTs determining their mechanical characteristics are considered. The applicability of the concepts of classical mechanics for describing the behavior of nanometer-sized objects is analyzed. The results of experimental investigations into the mechanical characteristics of CNTs and macroscopic objects containing nanotubes are discussed. The problem of reinforcement of polymer materials as a result of inserting carbon nanostructures is reviewed. Examples showing the effective operation of nanoelectromechanical systems (NEMSs) are analyzed.