Tissue engineering is a promising solution for the problem of organ or tissue shortage. A key requirement is the use of biologically functional scaffolds to deliver cells to the implant site and/or provide a structure for cell attachment to replace lost or damaged extracellular matrix. While a large number of polymers have been used to fabricate microscale woven and non-woven scaffolds for tissue regeneration, the natural extracellular matrix is nanoscale, a characteristic that should be incorporated into scaffold design for tissue engineering applications. A nanofibrous scaffold, such as those produced by the electrospinning process, has several unique advantages over microscale fibrous meshes, including high surface area to volume ratio, improved mechanical properties, and morphological similarities to natural extracellular matrix. Over the past five years, many polymers, including natural and synthetic, biodegradable and non-biodegradable, have been successfully electrospun into nanofibers, characterized, and evaluated for their potential applications for various tissue engineering and drug delivery applications. In this review, we survey these polymers and electrospun nanofibrous scaffolds and discuss the critical parameters for successful tissue engineering applications in the future.