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This paper presents a comprehensive critical review of technologies used in the development of fibre optic shape sensors (FOSSs). Their operation is based on multi-dimensional bend measurements using a series of fibre optic sensors. Optical fibre sensors have experienced tremendous growth from simple bend sensors in 1980s to full three-dimensional FOSSs using multicore fibres in recent years. Following a short review of conventional contact-based shape sensor technologies, the evolution trend and sensing principles of FOSSs are presented. This paper identifies the major optical fibre technologies used for shape sensing and provides an account of the challenges and emerging applications of FOSSs in various industries such as medical robotics, industrial robotics, aerospace and mining industry.