Tendon and ligament tissue regeneration and replacement are complex since scaffolds need to guarantee an adequate hierarchical structured morphology, and non-linear mechanical properties. Moreover, to guide the cells’ proliferation and tissue re-growth, scaffolds must provide a fibrous texture mimicking the typical of the arrangement of the collagen in the extracellular matrix of these tissues. Among the different techniques to produce scaffolds, electrospinning is one of the most promising, thanks to its ability to produce fibers of nanometric size. This manuscript aims to provide an overview to researchers approaching the field of repair and regeneration of tendons and ligaments. To clarify the general requirements of electrospun scaffolds, the first part of this manuscript presents a general overview concerning tendons’ and ligaments’ structure and mechanical properties. The different types of polymers, blends and particles most frequently used for tendon and ligament tissue engineering are summarized. Furthermore, the focus of the review is on describing the different possible electrospinning setups and processes to obtain different nanofibrous structures, such as mats, bundles, yarns and more complex hierarchical assemblies. Finally, an overview concerning how these technologies are exploited to produce electrospun scaffolds for tendon and ligament tissue applications is reported together with the main findings and outcomes.