There are a number of critical requirements for electrolytes in aqueous redox flow batteries. This paper reviews organic molecules that have been used as the redox-active electrolyte for the positive cell reaction in aqueous redox flow batteries. These organic compounds are centred around different organic redox active moieties such as the aminoxyl radical (TEMPO and N-hydroxyphthalimide), carbonyl (quinones and biphenols), amine (e.g indigo carmine), ether and thioether (e.g. thianthrene) groups. We consider the key metrics that can be used to assess their performance: redox potential, solubility, redox kinetics, diffusivity, operating pH, stability, and cost. We develop a new figure of merit - the theoretical intrinsic power density - which combines the first four of the aforementioned metrics to allow ranking of different redox couples on just one side of the battery. The organic electrolytes show theoretical intrinsic power densities which are 2-100 times larger than that of the VO2+/VO2+ couple, with TEMPO-derivatives showing the highest performance. Finally, we survey organic positive electrolytes in the literature on the basis of their redox-active moieties and the aforementioned figure of merit.