A first-order modified, hydrogenated polymer comprising (1) a hydrogenated polymer obtained by hydrogenating at least one unhydrogenated polymer selected from the group consisting of a polymer comprising conjugated diene monomer units and a copolymer comprising conjugated diene monomer units and vinyl aromatic hydrocarbon monomer units, and (2) a functional group-containing first-order modifier group bonded to the hydrogenated polymer (1), wherein the content of the vinyl aromatic hydrocarbon monomer units, vinyl aromatic hydrocarbon block ratio, weight average molecular weight, and hydrogenation ratio (as measured with respect to the double bonds in the conjugated diene monomer units) of the first-order modified, hydrogenated polymer are, respectively, within specific ranges. A second-order modified polymer obtained by reacting a second-order modifier with a first-order modified polymer comprising (.beta.) a base polymer and (.gamma.) a functional group-containing first-order modifier group bonded to the base polymer (.beta.). A composition comprising the above-mentioned first-order modified, hydrogenated polymer or second-order modified polymer and at least one functional component other than the modified polymer.