As worldwide energy consumption continues to increase, so too does the demand for improved energy storage technologies. Supercapacitors are energy storage devices that are receiving considerable interest due to their appealing features such as high power densities and much longer cycle lives than batteries. As such, supercapacitors fill the gaps between conventional capacitors and batteries, which are characterised by high power density and high energy density, respectively. Carbon nanomaterials, such as graphene nanoplatelets, are being widely explored as supercapacitor electrode materials due to their high surface area, low toxicity, and ability to tune properties for the desired application. In this review, we first briefly introduce the theoretical background and basic working principles of supercapacitors and then discuss the effects of electrode material selection and structure of carbon nanomaterials on the performances of supercapacitors. Finally, we highlight the recent advances of graphene nanoplatelets and how chemical functionalisation can affect and improve their supercapacitor performance.