Electrospinning has attracted tremendous interest in the research community as a simple and versatile technique to produce synthetic polymeric ultrafine fibres with diameters ranging from a few micrometres to tens of nanometres. Recently, some natural origin polymers have also been successfully electrospun. Owing to their very small diameter, polymeric nanofibres exhibit unusual properties such as high specific surface area, flexibility in surface functionalities and superior mechanical properties. In addition, electrospun non-woven meshes could physically mimic the extracellular matrix structure of native tissues. These remarkable properties render electrospun nanofibres useful for many applications, particularly those related to the field of biomedical engineering. The first part of this review is intended to provide a fundamental survey of the electrospinning process (apparatus, governing parameters) and of recent improvements of the technique, including associated structural modifications of polymeric nanofibre meshes. The prospective tissue engineering/biomedical applications of electrospun polymeric nanofibres are then reviewed, namely, wound dressings, medical prostheses, drug delivery systems, DNA release and tissue engineering scaffolds. The essential properties of scaffolds in terms of the structural features of electrospun nanofibre meshes are discussed. Finally, the future perspectives for applications of electrospun nanofibres, particularly in the field of tissue engineering, are considered.