Over the past decade, there has been significant growth in the use of fibre-based products and structures for biomedical applications, with new fibrous materials and manufacturing technologies being introduced in various medical sectors. Wet spinning is one of the most common manufacturing techniques for the fabrication of fibrous structures. Novel wet spinning-based fabrication methods, such as computer-aided wet spinning, rotary wet spinning, microfluidic wet spinning, and channel-based wet spinning techniques, have been developed to create bespoke fibrous materials for medical use. A wide range of fibrous materials from natural and synthetic polymers are now available with various designs and properties, including core-shell fibres, hierarchical structures, hollow fibres, and braided design. The present review provides a detailed overview of the wet spinning process, the effects of process parameters on the surface texture of wet-spun fibres, and novel wet spinning-based manufacturing techniques. Various wet-spun structures and materials and the cellular response to the surface characteristics of wet-spun fibres are discussed. An overview is also given on the potential applications of wet-spun fibrous structures in cardiac, bone, tendon, ligament, and nerve repair, annulus fibrosus regeneration, shape memory surgical sutures, wound healing, drug delivery, and hemodialysis systems.