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A synthesis of recent progress is presented on a topic that lies at the heart of both structural engineering and non-linear science. The emphasis is on thin elastic structures that lose stability subcritically—without a nearby stable post-buckled state—a canonical example being a uniformly axially-loaded cylindrical shell. Such structures are hard to design and certify because imperfections or shocks trigger buckling at loads well below the threshold of linear stability. A resurgence of interest in structural instability phenomena suggests practical stability assessments require stochastic approaches and imperfection maps. This article surveys a different philosophy; the buckling process and ultimate post-buckled state are well-described by the perfect problem. The significance of the Maxwell load is emphasized, where energy of the unbuckled and fully-developed buckle patterns are equal, as is the energetic preference of localized states, stable, and unstable versions of which connect in a snaking load-deflection path. The state of the art is presented on analytical, numerical and experimental methods. Pseudo-arclength continuation (path-following) of a finite-element approximation computes families of complex localized states. Numerical implementation of a mountain-pass energy method then predicts the energy barrier through which the buckling process occurs. Recent developments also indicate how such procedures can be replicated experimentally; unstable states being accessed by careful control of constraints, and stability margins assessed by shock sensitivity experiments. Finally, the fact that subcritical instabilities can be robust, not being …