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In this study, the mechanical properties of epoxy nanocomposites with graphene platelets, single-walled carbon nanotubes, and multi-walled carbon nanotube additives were compared at a nanofiller weight fraction of 0.1 +/- 0.002%. The mechanical properties measured were the Young's modulus, ultimate tensile strength, fracture toughness, fracture energy, and the material's resistance to fatigue crack propagation. The results indicate that graphene platelets significantly out-perform carbon nanotube additives. The Young's modulus of the graphene nanocomposite was approximately 31% greater than the pristine epoxy as compared to approximately 3% increase for single-walled carbon nanotubes. The tensile strength of the baseline epoxy was enhanced by approximately 40% with graphene platelets compared to approximately 14% improvement for multi-walled carbon nanotubes. The mode I fracture toughness of the nanocomposite with graphene platelets showed approximately 53% increase over the epoxy compared to approximately 20% improvement for multi-walled carbon nanotubes. The fatigue resistance results also showed significantly different trends. While the fatigue suppression response of nanotube/epoxy composites degrades dramatically as the stress intensity factor amplitude is increased, the reverse effect is seen for graphene-based nanocomposites. The superiority of graphene platelets over carbon nanotubes in terms of mechanical properties enhancement may be related to their high specific surface area, enhanced nanofiller-matrix adhesion/interlocking arising from their wrinkled (rough) surface, as well as the two-dimensional (planar) geometry of graphene platelets.