Tissue engineering is an emergent and very interesting research field, providing potential solutions for a myriad of challenges in healthcare. Fibrous scaffolds specifically have shown promise as an effective tissue engineering method, as their high length-to-width ratio mimics that of extracellular matrix components, which in turn guides tissue formation, promotes cellular adhesion and improves mechanical properties. In this review paper, we discuss in detail both the importance of fibrous scaffolds for the promotion of tissue growth and the different methods to produce fibrous biomaterials to possess favorable and unique characteristics. Here, we focus on the pressing need to develop biomimetic structures that promote an ideal environment to encourage tissue formation. In addition, we discuss different biomedical applications in which fibrous scaffolds can be useful, identifying their importance, relevant aspects, and remaining significant challenges. In conclusion, we provide comments on the future direction of fibrous scaffolds and the best way to produce them, proposed in light of recent technological advances and the newest and most promising fabrication techniques.