Buckling and post-buckling have been expensively considered as critical limit states of slender/thin members over the centuries. Recent research interests have been shifted to exploit the potential "smart applications" of buckling-induced elastic instability at multiscale. For example, buckling elements have been implemented to develop monostable, bistable and multistable mechanisms for different functional utilities. To design buckling-induced techniques and optimize outputs, it is of necessity to accurately control the mechanical response of buckling elements. This review paper aims at showcasing the theoretical studies of multiscale buckling and post-buckling analysis mainly conducted during the last two decades. The studies of micro/nanoscale buckling and post-buckling analysis are discussed with respect to the nonlocal elasticity, non-classical couple stress and strain gradient elasticity theories, respectively, while the macroscale buckling analysis is categorized into the small and large deformation studies. A comparison is presented between the small and large deformation models at macroscale to identify the applicability of the theories. Recent research trends are reviewed on the optimal design of buckling-induced mechanisms and techniques. In the end, we discuss the potential research avenues for future innovations.