A composite oxide catalyst for the oxidation of an olefin containing Mo and Bi as essential components, characterized in that it has a specific surface area of 5 to 25 m.sup.2/g and a pore volume of 0.2 to 0.7 cc/g, and has a pore diameter distribution wherein the volume of the pores having a pore diameter of 0.03 to 0.1 .mu.m accounts for 30% or more of the total pore volume, the volume of the pores having a pore diameter of 0.1 to 1 .mu.m accounts for 20% or more of the total pore volume, and the volume of the pores having a pore diameter of less than 0.03 .mu.m is 10% or less of the total pore volume; a composite oxide catalyst for use in the vapor phase catalytic oxidation of acrolein or methacrolein or the like which comprises Mo, Bi and a halogen; either of the above two composite oxide catalysts which comprises Mo, Bi, Fe, Si and an element selected from alkali metals and thallium, and optionally Co, Ni, Mg, Ca, Zn, Ce, Sm, a halogen, B, P, As and W; a method for preparing any of the above composite catalysts; and a method for using any of the above composite oxide catalysts.