This chapter deals with the elastic postbuckling behavior of structural systems, beginning with an overview of structural stability fundamentals and later the critical points, bifurcation, and limit points. Using the total potential energy (TPE) framework, various postbuckling paths encountered in stability analysis are discussed. A review of the secant matrix technique and its applications across different structural types, such as trusses, beams, and thin-walled sections, are presented. The chapter also covers computational methods, focusing on iterative and incremental approaches, particularly the arc-length method for path following in postbuckling analysis. In summary, the computational and analytical bases are presented for a deep understanding of the postbuckling response of structures in practical applications.