An effective therapeutic strategy to treat tendon or ligament injury continues to be a clinical challenge due to the limited natural healing capacity of these tissues. Furthermore, the repaired tendons or ligaments usually possess inferior mechanical properties and impaired functions. Tissue engineering can restore the physiological functions of tissues using biomaterials, cells, and suitable biochemical signals. It has produced encouraging clinical outcomes, forming tendon or ligament-like tissues with similar compositional, structural, and functional attributes to the native tissues. This paper starts by reviewing tendon/ligament structure and healing mechanisms, followed by describing the bioactive nanostructured scaffolds used in tendon and ligament tissue engineering, with emphasis on electrospun fibrous scaffolds. The natural and synthetic polymers for scaffold preparation, as well as the biological and physical cues offered by incorporating growth factors in the scaffolds or by dynamic cyclic stretching of the scaffolds, are also covered. It is expected to present a comprehensive clinical, biological, and biomaterial insight into advanced tissue engineering-based therapeutics for tendon and ligament repair.