Representing a new class of nanoscale material, carbon nanotubes possess many extraordinary mechanical and electronic properties stemming essentially from their unique geometric and chemical structures. Through more than two decades of extensive theoretical and experimental investigations, our understanding on the mechanical properties of carbon nanotubes has greatly improved. The intrinsic mechanical properties of carbon nanotubes, such as their stiffness, strength and deformability, have been relatively well studied and understood; and other mechanics-related properties of carbon nanotubes, such as the defect formation, the fracture mechanism, the interface mechanics and the electromechanics, have also being broadly examined and a comprehensive knowledge of them begins to emerge. I review the current status of research on the mechanical study of carbon nanotubes, especially on the experimental study of their fundamental mechanical properties, such as Young’s modulus, tensile and shear strength, compressibility and deformability. Selected experimental methods and techniques used for the studies will also be introduced. I conclude the review by discussing the new challenges still facing the mechanical study of carbon nanotubes.