The present invention relates to a supercapacitor, also known as an electrical double-layer capacitor or ultracapacitor, having electrode material comprising single-wall carbon nanotubes. The carbon nanotubes can be derivatized with functional groups. The electrode material is made by preparing a polymer-nanotube suspension comprising polymer and nanotubes, forming the polymer-nanotube suspension into a polymer-nanotube composite of the desired form, carbonizing the polymer-nanotube composite to form a carbonaceous polymer-nanotube material, and activating the material. The supercapacitor includes electrode material comprising activated carbonaceous polymer-nanotube material in contact with current collectors and permeated with an electrolyte, which may be either fluid or solid. In the case of a fluid or compressible electrolyte, an electrolyte-permeable separator or spacer is interposed between the electrodes to keep the electrodes from shorting. The supercapacitor made with electrodes comprising underivatized single-wall carbon nanotubes and polymer that has been carbonized and activated appears to operate as a non-Faradaic supercapacitor.