Polymer structures are essential in biomedical applications, such as tissue engineering and regenerative medicine, due to their properties of biocompatibility and biodegradability. The use of polymer composites is particularly significant for various applications, as a combination of mechanical, chemical and biological properties can be achieved. The intended application determines the selection of the biopolymer materials. In addition, the method used to process the materials should produce scaffolds that deliver the required therapeutic effect. Electrospinning is a well-established technique for producing fibres from a spectrum of polymer feedstock with diameters ranging from micrometres to nanometres. Moreover, electrospun fibres may exhibit a variety of biomimetic surface features (such as porosity) and orientations. The present review is aimed at describing the synthetic and natural polymers commonly used in biomedical applications, with particular focus on composite polymers. The processing of polymer composites using various electrospinning techniques is also discussed, as well as the biomedical application of the electrospun polymer composites. Although polymer composites and electrospinning have seen significant growth in biomedicine, making them suitable candidates for Industry 4.0, recent advances have shown great potential in propelling them into Industry 5.0 applications. These advances include the fabrication of smart 3D multi-functional scaffolds with tailored mechanical and biological properties, and the inclusion of machine learning technologies.