Electrospinning has been employed extensively in tissue engineering to generate nanofibrous scaffolds from either natural or synthetic biodegradable polymers to simulate the cellular microenvironment. Electrospinning rapidly produces fibers of the nanolength scale and the process offers many opportunities to tailor the physical, chemical, and biological properties of a material for specific applications and cellular environments. There is growing evidence that nanofibers amplify certain biological responses such as contact guidance and differentiation, however this has not been fully exploited in tissue engineering. This review addresses the cellular interactions with electrospun scaffolds, with particular focus on neural, bone, cartilage, and vascular tissue regeneration. Some aspects of scaffold design, including architectural properties, surface functionalization and materials selection are also addressed.