Almost every form of alternative energy and energy system being implemented today, e.g., wind, solar, hybrid electric and hydrogen fuel cell vehicles, depends on electrical energy storage (EES). At the DOE basic energy sciences workshop on Basic Research Needs for Electrical Energy Storage held April, 2007, the conclusion was reached that "revolutionary breakthroughs in EES are perhaps the most crucial need for this nation’s secure energy future." The workshop, chaired by John Goodenough, University of Texas-Austin, focused on the two primary methods of EES - batteries and electrochemical double-layer capacitors (also referred to as ‘ultracapacitors’ and ‘supercapacitors’). As stated in the report from this DOE workshop, “The performance of current EES technologies falls well short of requirements for using electrical energy efficiently in transportation, commercial, and residential applications.” In this dissertation, increasing the energy storage capacity of ultracapacitors through the use of graphene electrode materials is investigated. Chapter 1 is a basic overview of EES applications and ultracapacitor technology. In Chapter 2, best practice experimental procedures to accurately evaluate a material’s performance are described. Because current measurement methods for determining a material’s performance for use as an ultracapacitor electrode are not well standardized, the different techniques currently being employed lead to wide variations in reported results. Reliable methods that would accurately test a large number of samples involving minute quantities of material were required. In Chapter 3, the performance of graphene-derived materials is investigated. Chemically modified graphene materials gave values competitive with current activated carbons and an ultracapacitor based on activated graphene electrodes yielded the highest specific capacitance values reported to date. Chapter 4 describes a lithium ion hybrid supercapacitor using this novel material that gave energy densities greater than lead acid batteries. The exceptional performance of these graphene derived materials will likely result in their rapid adoption as well as an expanded range of applications utilizing ultracapacitors. As increasingly higher surface area graphene materials are developed, a fundamental understanding of the components that affect interfacial capacitance is critical for further capacity increases. In the last chapter, the first direct measurement of the interfacial capacitance for one and two sides of single layer graphene is presented. The results show that the quantum capacitance increasingly becomes a factor with the result being a reduced increase in capacitance, not the linear increase with surface area as would be expected for bulk conductive materials. These results indicate that the development of higher surface area graphene materials alone is not sufficient for additional increases in performance; the modification of the electronic properties will also be required. / text