Nanofibrous biomaterials have huge potential for drug delivery, due to their structural features and functions that are similar to the native extracellular matrix (ECM). A wide range of natural and polymeric materials can be employed to produce nanofibrous biomaterials. This review introduces the major natural and synthetic biomaterials for production of nanofibers that are biocompatible and biodegradable. Different technologies and their corresponding advantages and disadvantages for manufacturing nanofibrous biomaterials for drug delivery were also reported. The morphologies and structures of nanofibers can be tailor-designed and processed by carefully selecting suitable biomaterials and fabrication methods, while the functionality of nanofibrous biomaterials can be improved by modifying the surface. The loading and releasing of drug molecules, which play a significant role in the effectiveness of drug delivery, are also surveyed. This review provides insight into the fabrication of functional polymeric nanofibers for drug delivery.