NOVELTY - Producing hydrogen and oxygen, involves: (a1) oxidizing mediator at working electrode to yield oxidized mediator, and reducing protons at counter electrode to yield hydrogen; and (b1) reducing oxidized mediator at working electrode to yield mediator, and oxidizing water at counter electrode to yield oxygen. The oxygen generation step is performed non-simultaneously to hydrogen generation step, and the oxidized mediator of step (a1) is used as oxidized mediator of step (b1), and the mediator has a reversible redox wave lying between onset of oxygen and hydrogen evolution reactions.    USE - For producing hydrogen and oxygen (claimed).    ADVANTAGE - The present methods have at least 90% Faradaic efficiency, and have substantially 100% Faradaic efficiency. The mediator is suitable for use in methods having a high current flow. The use of the mediator is not associated with mass transfer limitations in the electrochemical reactions. The rate of electron and/or proton transfer is relatively fast, and useful quantities of hydrogen and oxygen may be produced in a practical time frame. Also the mediator has the ability to act as a pH buffer to the electrolyte by accepting protons that are generated in an oxygen production method. Conversely, as protons are consumed in a hydrogen generation method, the mediator is capable of donating protons to the system.    DETAILED DESCRIPTION - Producing hydrogen and oxygen, involves: (a1) oxidizing a mediator at a working electrode to yield an oxidized mediator, and reducing protons at a counter electrode to yield hydrogen; and (b1) reducing an oxidized mediator at a working electrode to yield a mediator, and oxidizing water at a counter electrode to yield oxygen. The oxygen generation step is performed non-simultaneously to the hydrogen generation step, and the oxidized mediator of step (a1) is used as the oxidized mediator of step (b1), or the mediator of step (b1) is used as the mediator of step (a1), and the mediator has a reversible redox wave lying between the onset of the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER). INDEPENDENT CLAIMS are included for the following:    (1) an electrochemical cell comprising a working electrode and a counter electrode, optionally together with a reference electrode, where the working electrode and the counter electrode define an electrolyte space, which space is divided into a working electrode space and a counter electrode space by a semi-permeable membrane; and further comprising an aqueous electrolyte within the electrolyte space, where the aqueous electrolyte in the working electrode space of the electrolyte space contains a mediator, and the semi-permeable membrane is impermeable to the mediator, and the mediator has a reversible redox wave lying between the onset of the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER);    (2) producing hydrogen, involving oxidizing a mediator at a working electrode to yield an oxidized mediator, and reducing protons at a counter electrode to yield hydrogen, and the mediator has a reversible redox wave lying between the onset of the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER);    (3) producing oxygen, involving reducing an oxidized mediator at a working electrode to yield a mediator, and oxidizing water at a counter electrode to yield oxygen and the mediator has a reversible redox wave lying between the onset of the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER);    (4) use (m1) of a polyoxometallate as an electron acceptor in a method of generating oxygen from water and/or the use of a polyoxometallate as an electron donor in a method of generating hydrogen from water; and    (5) use (m2) of a quinone compound as an electron acceptor in a method of generating oxygen from water and/or the use of a quinone compound as an electron donor in a method of generating hydrogen from water, where the compound is a quinone in its oxidized form.