As the fraction of electricity generation from intermittent renewable sources--such as solar or wind--grows, the ability to store large amounts of electrical energy is of increasing importance. Solid-electrode batteries maintain discharge at peak power for far too short a time to fully regulate wind or solar power output. In contrast, flow batteries can independently scale the power (electrode area) and energy (arbitrarily large storage volume) components of the system by maintaining all of the electro-active species in fluid form. Wide-scale utilization of flow batteries is, however, limited by the abundance and cost of these materials, particularly those using redox-active metals and precious-metal electrocatalysts. Here we describe a class of energy storage materials that exploits the favourable chemical and electrochemical properties of a family of molecules known as quinones. The example we demonstrate is a metal-free flow battery based on the redox chemistry of 9,10-anthraquinone-2,7-disulphonic acid (AQDS). AQDS undergoes extremely rapid and reversible two-electron two-proton reduction on a glassy carbon electrode in sulphuric acid. An aqueous flow battery with inexpensive carbon electrodes, combining the quinone/hydroquinone couple with the Br2/Br- redox couple, yields a peak galvanic power density exceeding 0.6Wcm-2 at 1.3Acm-2. Cycling of this quinone-bromide flow battery showed >99 per cent storage capacity retention per cycle. The organic anthraquinone species can be synthesized from inexpensive commodity chemicals. This organic approach permits tuning of important properties such as the reduction potential and solubility by adding functional groups: for example, we demonstrate that the addition of two hydroxy groups to AQDS increases the open circuit potential of the cell by 11% and we describe a pathway for further increases in cell voltage. The use of π-aromatic redox-active organic molecules instead of redox-active metals represents a new and promising direction for realizing massive electrical energy storage at greatly reduced cost.