Traditionally, structures have been designed to work below critical loads, while attainment of an instability was normally identified as connected to failure or, at least, to loss of functionality of the involved structural elements. Therefore, structural deformations under service loads were small, so that instability and bifurcation were viewed simply as potentially dangerous phenomena, a perspective that can be summarized in the word ‘buckliphobia’. Recently, a new research stimulus has derived from the observation that soft structures, such as for instance biological systems, but also rubber and gel, may work in a post-critical regime, where elastic elements are subject to extreme deformations, though still exhibiting excellent mechanical performances. The possibility of exploiting highly deformable structures opens new and unexpected technological possibilities. In particular, the challenge is the design of deformable and bi-stable mechanisms which can reach superior mechanical performances and can have a strong impact on several high-tech applications, including stretchable electronics, nanotube serpentines, deployable structures for aerospace engineering, cable deployment in the ocean, but also sensors and flexible actuators and vibration absorbers. The so-called ‘extreme mechanics’ is an emerging branch of instability of solids and structures aimed at the investigation of instabilities as related to pattern formation and the subsequent large deformation nonlinear behavior, a design approach that can be summarized with the sentence ‘joy of buckling’. We draw the attention on recent results on how to exploit the post-critical path of an elastic structure to obtain flexible mechanisms with special behaviours: (i.) a spherical shell shrinking towards its center, a problem related to buckling of periodic structures; (ii.) a one-degree-of-freedom elastic structure buckling in tension and compression and providing a constant force (‘neutral’) post-critical behaviour; (iii.) dynamical instabilities explaining wrapping of a liquid drop by an elastic strip; (iv.) wrinkling of thin films attached to a soft substrate. Participants will be introduced to a variety of interrelated topics involving the mechanics of extremely deformable structures, with emphasis on bifurcation, instability and nonlinear behaviour, both in the quasi-static and dynamic regimes. Essential and up-to-date theoretical, numerical, and experimental methodologies will be covered, as a tool to progress towards a satisfactory modelling of the nonlinear behaviour of structures. In this way, the course will provide a unique opportunity to learn simultaneously a broad range of subjects and techniques that are a prerequisite to research in the fields of highly deformable structures and thin films, and to the design of deformable mechanisms, adaptive and periodic structures, and stretchable electronics. Finally, it will be shown how the mechanics of highly deformable structures is the key to understanding several phenomena in biomechanics, such as morphogenesis, growth and propulsion.