Carbon nanotubes have mechanical properties that are far in excess of conventional fibrous materials used in engineering polymer composites. Effective reinforcement of polymers using carbon nanotubes is difficult due to poor dispersion and alignment of the nanotubes along the same axis as the applied force during composite loading. This paper reviews the mechanical properties of carbon nanotubes and their polymer composites to highlight how many previously prepared composites do not effectively use the excellent mechanical behaviour of the reinforcement. Nanomechanical tests using atomic force microscopy are carried out on simple uniaxially aligned carbon nanotube-reinforced polyvinyl alcohol (PVA) fibres prepared using electrospinning processes. Dispersion of the carbon nanotubes within the polymer is achieved using a surfactant. Young's modulus of these simple composites is shown to approach theoretically predicted values, indicating that the carbon nanotubes are effective reinforcements. However, the use of dispersant is also shown to lower Young's modulus of the electrospun PVA fibres.