NOVELTY - A nickel-cobalt-manganese composite hydroxide (I) has specific surface area of 1-10 m2/g as measured by nitrogen adsorption BET method, carbon content of 0.1 %mass or less as measured by high-frequency infrared combustion method is 0.1 mass % or less, and half-power band width of 1.5 degrees or less in (101) plane as measured by X-ray diffraction method.    USE - Nickel-cobalt-manganese composite hydroxide is used as precursor of positive electrode active material of non-aqueous electrolyte secondary battery. Uses include but are not limited to mobile telephone, notebook-type personal computer, electric motor vehicle and hybrid motor vehicle.    ADVANTAGE - The nickel-cobalt-manganese composite hydroxide has high crystallinity and thermal stability. The non-aqueous electrolyte secondary battery containing nickel-cobalt-manganese composite hydroxide has excellent cycle characteristics, high capacity and improved safety.    DETAILED DESCRIPTION - A nickel-cobalt-manganese composite hydroxide of formula: Ni1-x-y-zCoxMnyMz(OH)2 (I) has specific surface area of 1-10 m2/g measured by nitrogen adsorption BET method, carbon content of 0.1 %mass or less as measured by high-frequency infrared combustion method is 0.1 mass % or less, and half-power band width of 1.5 degrees or less in (101) plane as measured by X-ray diffraction method.    x,y=0-1/3;    z=0-0.1;and    M= magnesium, aluminum, calcium, titanium, vanadium, chromium, zirconium, niobium, molybdenum or tungsten.    An INDEPENDENT CLAIM is included for manufacture of nickel-cobalt-manganese composite hydroxide, which involves supplying aqueous solution containing nickel salt, cobalt salt and/or manganese salt, and aqueous solution containing ammonium-ion supply material to a reaction tank, mixing, adding caustic-alkali aqueous solution to resultant mixture, adjusting pH to 11-13, reacting at 25 degrees C, crystallizing nickel-cobalt-manganese composite hydroxide particles in the obtained reaction solution, carrying out solid-liquid-separation process, washing crystallized nickel-cobalt-manganese composite hydroxide particles using water, and drying. The oxygen concentration of the atmosphere in the reaction tank which contacts the open surface of the reaction solution is 0.2 vol.% or less.