Among the objects of nanomechanics studies, carbon nanotubes have attracted special interest due to their unique properties and potential use in a wide range of applications, including nanometer-scale devices and composite materials. In these applications the mechanical responses of the nanotubes can significantly affect the performance of the devices and materials. Therefore, characterizing and understanding their mechanical responses is necessary in order to optimize their utilization in these applications. Computational simulations are uniquely able to provide insights that are challenging to obtain experimentally. Molecular dynamics simulations in particular are popular for the examination of the mechanical responses of nanotubes. This chapter provides a review of the background of molecular dynamics simulation methods, their role in the study of the nanomechanical responses of carbon nanotubes, and their important contributions to this emerging research field. Illustrative examples are presented that illustrate how these approaches are providing new and exciting insights into nanomechanical properties as elastic modulus or stiffness, fracture, and bucklng. Furthermore, the simulations indicate that filling or functionalization, combined loads, and external gases influence these properties. Thus, molecular dynamics simulation methods are revolutionizing our understanding of the mechanical behavior of nanotube systems at the most fundamental atomic level.