The invention relates to a catalyst for decomposing nitrous oxide, which is [1] a catalyst comprising a support having supported thereon aluminum, magnesium and rhodium, [2] a catalyst comprising an alumina support having supported thereon magnesium and rhodium, [3] a catalyst comprising a support having supported thereon rhodium, the support comprising a spinel crystalline composite oxide formed by magnesium and at least a part of aluminum, [4] a catalyst comprising a support having supported thereon aluminum, rhodium and at least one metal selected from zinc, iron, manganese and nickel, [5] a catalyst comprising an alumina support having supported thereon rhodium and at least one metal selected from zinc, iron, manganese and nickel, or [6] a catalyst comprising a support having supported thereon rhodium, the support comprising a spinel crystalline composite oxide formed by at least a part of aluminum and the at least one metal selected from zinc, iron, manganese and nickel. The catalyst is not easily deteriorated in the activity due to moisture, favored with low-temperature decomposition activity and capable of reducing the amount of NOx generated to the allowable concentration or less. The invention also relates to a process for producing the catalyst and to a method for decomposing nitrous oxide.