NOVELTY - Manufacture of positive electrode active material involves using positive electrode active material contains lithium-transition metal composite oxide (I), and comprised by particles of polycrystalline structure. A niobium salt solution and an acid are added to slurry of a nickel-containing hydroxide, and nickel-containing hydroxide which is coated with niobium compound is obtained. The obtained product is mixed with a lithium compound, and a lithium mixture is obtained. The lithium mixture is baked, to obtain lithium-transition metal composite oxide (I).    USE - Manufacture of positive electrode active material used for non-aqueous electrolyte secondary batteries (claimed) used as power supply of large-sized secondary battery for hybrid vehicles and electric vehicle, and fixed-type storage batteries for portable electronic device e.g. mobile telephone and notebook computer.    ADVANTAGE - The method enables easy manufacture of positive electrode active material having high thermal-stability and charge/discharge capacitance, and provides economical and safe non-aqueous electrolyte secondary battery having excellent cycling characteristics and durability.    DETAILED DESCRIPTION - Manufacture of positive electrode active material involves using positive electrode active material contains lithium-transition metal composite oxide of formula: LidNi1-a-b-cCoaMbNbcO2 (I), and comprised by particles of polycrystalline structure. A niobium salt solution and an acid are simultaneously added to slurry of a nickel-containing hydroxide, pH of the slurry is controlled to 7-11 at reference temperature of 25 degrees C, and nickel-containing hydroxide which is coated with niobium compound is obtained. The nickel-containing hydroxide coated with the niobium compound is mixed with a lithium compound, and a lithium mixture is obtained. The lithium mixture is baked at 700-830 degrees C in an oxidizing atmosphere, to obtain lithium-transition metal composite oxide (I) contained in the positive electrode active material.    M=Mn, V, Mg, Ti or Al;    a=0.05-0.35;    b=0-0.1;    c=0.006-0.06;and    d=0.95-1.2.    INDEPENDENT CLAIMS are included for the following:    (1) positive electrode active material, which contains the lithium-transition metal composite oxide (I), and has porous structure and specific surface area of 2-7 m2/g; and    (2) non-aqueous electrolyte secondary battery, which contains the positive electrode active material used for a positive electrode.