The unique and exceptional physical properties of carbon nanotubes have inspired their use as a filler within a polymeric matrix to produce carbon nanotube polymer composites with enhanced mechanical, thermal and electrical properties. A powerful method of synthesising nanofibers comprising these polymer composites is electrospinning, which utilises an applied electric stress to draw out a thin nanometer-dimension fiber from the tip of a sharp conical meniscus. The focussing of the flow due to converging streamlines at the cone vertex then ensures alignment of the carbon nanotubes along the fiber axis, thus enabling the anisotropic properties of the nanotubes to be exploited. We consider the work that has been carried out to date on various aspects encompassing preprocessing, synthesis and characterisation of these electrospun polymer composite nanofibers as well as the governing mechanisms and associated properties of such fibers. Particular attention is also dedicated to the theoretical modelling of these fiber systems, in particular to the electrohydrodynamic modelling of electrospinning polymer jets.