The nerves of the peripheral nervous system are not able to effectively regenerate in cases of severe neural injury. This can result in debilitating consequences, including morbidity and lifelong impairments affecting the quality of the patient's life. Recent findings in neural tissue engineering have opened promising avenues to apply fibrous tissue-engineered scaffolds to promote tissue regeneration and functional recovery. These scaffolds, known as neural scaffolds, are able to improve neural regeneration by playing two major roles, namely, by being a carrier for transplanted peripheral nervous system cells or biological cues and by providing structural support to direct growing nerve fibers towards the target area. However, successful implementation of scaffold-based therapeutic approaches calls for an appropriate design of the neural scaffold structure that is capable of up- and down-regulation of neuron-scaffold interactions in the extracellular matrix environment. This review discusses the main challenges that need to be addressed to develop and apply fibrous tissue-engineered scaffolds in clinical practice. It describes some promising solutions that, so far, have shown to promote neural cell adhesion and growth and a potential to repair peripheral nervous system injuries.