This review explains how an understanding of relationships between the fine structure of fibres and their physical properties has developed over the past 60 or 70 years, culminating in the production of high-tech fibres with greatly superior properties, utilised in technical textiles, which are now commonplace aspects of modern life. In the early days, when natural fibres of cellulose were first studied by X-ray-diffraction methods, the idea of long-chain molecules passing through both crystalline and disordered regions of structure was proposed and generally accepted, but it was only with the invention of electron microscopy, and its application to fibre science, that a true understanding of the diversity of fibre structure could be appreciated and related to physical properties. Thus, through electron diffraction, it was realised that the long-chain molecules of the synthetic fibres of polyethylene, polyamide, and polyester had a propensity to fold into lamellar crystals perpendicular to the fibre axis, which provided a severe restriction to the development of high-performance fibres. The search for chain-extended polymers that could he spun into fibres was ongoing, and simultaneously the possibility of producing fibres with extended lamellar systems parallel to the fibre axis, by converting precursor fibres into carbon-layer planes, was being explored. Direct visualisation of the molecular structures of PPTA and carbon fibres in transmission electron microscopy led to improved understanding of the relationships between tensile properties and structure, together with a realisation that the concomitant problem of relatively poor compressive properties has yet to be overcome.