Achieved is a nickel-cobalt-manganese composite hydroxide which is excellent in reactivity with a lithium compound, and able to achieve a positive electrode active material which has excellent thermal stability and battery characteristics. The nickel-cobalt-manganese composite hydroxide is intended to serve as a precursor for a positive electrode active material of a non-aqueous electrolyte secondary battery, and represented by a general formula: Ni1-x-y-zCoxMnyMz(OH)2 (0