One-dimensional (1-D) nanomaterials exhibit great potentials in their applications to functional materials, nano-devices and systems owing to their excellent properties. In the past decade, considerable studies have been done, with new patents being developed, on these 1-D building blocks for for their mechanical properties, especially elastic properties, which provide a solid foundation for the design of nanoelectromechanical systems (NEMS) and predictions of reliability and longevity for their devices. This paper reviews some of the recent investigations on techniques as well as patents available for the quantitative characterization of the elastic behaviors of various 1-D nanomaterials, with particular focus on on-chip testing system. The review begins with an overview of major testing methods for 1-D nanostructures' elastic properties, including nanoindentation testing, AFM (atomic force microscopy) testing, in situ SEM (scanning electron microscopy) testing, in situ TEM (transmission electron microscopy) testing and the testing system on the basis of MEMS (micro-electro-mechanical systems) technology, followed by advantages and challenges of each testing approach. This review also focuses on the MEMS-based testing apparatus, which can be actuated and measured inside SEM and TEM with ease, allowing users to highly magnify the continuous images of the specimen while measuring load electronically and independently. The combination of on-chip technologies and the in situ electron microscopy is expected to be a potential testing technique for nanomechanics. Finally, details are presented on the key challenges and possible solutions in the implementation of the testing techniques referred above.