Tissue engineering (TE) is an emerging field of study that incorporates the principles of biology, medicine, and engineering for designing biological substitutes to maintain, restore, or improve tissue functions with the goal of avoiding organ transplantation. Amongst the various scaffolding techniques, electrospinning is one of the most widely used techniques to synthesise a nanofibrous scaffold. Electrospinning as a potential tissue engineering scaffolding technique has attracted a great deal of interest and has been widely discussed in many studies. The high surface-to-volume ratio of nanofibres, coupled with their ability to fabricate scaffolds that may mimic extracellular matrices, facilitates cell migration, proliferation, adhesion, and differentiation. These are all very desirable properties for TE applications. However, despite its widespread use and distinct advantages, electrospun scaffolds suffer from two major practical limitations: poor cell penetration and poor load-bearing applications. Furthermore, electrospun scaffolds have low mechanical strength. Several solutions have been offered by various research groups to overcome these limitations. This review provides an overview of the electrospinning techniques used to synthesise nanofibres for TE applications. In addition, we describe current research on nanofibre fabrication and characterisation, including the main limitations of electrospinning and some possible solutions to overcome these limitations.