A modified block copolymer composition comprising: a modified block copolymer consisting of a polymer block A comprising primarily a vinyl aromatic hydrocarbon and a polymer block B comprising primarily a conjugated diene, or a hydrogenation product of the copolymer, and at least one of fillers selected from the group consisting of silica-based inorganic fillers, metal oxides and metal hydroxides, wherein the molecular chain of the component has at the terminals thereof a functional group selected from the group consisting of the groups having the following formulae to, and: -NR 9 -R 10 -OH -N[R 10 -OH] 2 where R 9, R 12 to R 14 are a hydrogen, a hydrocarbon group having carbon atoms of 1 to 24, or a hydrocarbon group having a functional group selected from the group consisting of a hydroxide group, an epoxy group, a silanol group, and an alkoxysilane group and having carbon atoms of 1 to 24; R 10 is a hydrocarbon chain having carbon atoms of 1 to 30, or a hydrocarbon chain having a functional group selected from the group consisting of a hydroxide group, an epoxy group, a silanol group, and an alkoxysilane group and having carbon atoms of 1 to 30; hydrocarbon groups R 9, R 12 to R 14 and a hydrocarbon chain R 10 may have an element such as oxygen, nitrogen or silicon bonded in the bonding form other than the hydroxide group, the epoxy group, the silanol group, and the alkoxysilane group; and R 11 is a hydrogen or an alkyl group having carbon atoms of 1 to 8; the component has a content of the vinyl aromatic hydrocarbon of 5 to 95 % by weight; the amount of the component is 0.5 to 50 parts by weight based on 100 parts by weight of the component; and the component has an average particle size in a dispersion of 0.01 to 2 μm, wherein the modified block copolymer has a structure expressed by one of the following general formulae: (A-B) n -X, (B-A) n -X, A-(B-A) n -X, B-(A-B) n -X, X-(A-B) n -X, X-A-(B-A) n -X, X-B-(A-B) n -X, [(B-A)n] m -X, [(A-B)n] m -X, [(B-A)n-B] m -X, [(A-B)n-A] m -X where A represents a polymer block comprising primarily a vinyl aromatic hydrocarbon, and B represents a polymer block comprising primarily a conjugated diene, n is an integer of 1 or more, generally 1 to 5, and m is an integer of 2 or more, generally 2 to 10, and X is a modifying agent's residue having the functional group selected from to, and bonded to the terminals of its molecular chain. a modified block copolymer consisting of a polymer block A comprising primarily a vinyl aromatic hydrocarbon and a polymer block B comprising primarily a conjugated diene, or a hydrogenation product of the copolymer, and at least one of fillers selected from the group consisting of silica-based inorganic fillers, metal oxides and metal hydroxides, R 9, R 12 to R 14 are a hydrogen, a hydrocarbon group having carbon atoms of 1 to 24, or a hydrocarbon group having a functional group selected from the group consisting of a hydroxide group, an epoxy group, a silanol group, and an alkoxysilane group and having carbon atoms of 1 to 24; R 10 is a hydrocarbon chain having carbon atoms of 1 to 30, or a hydrocarbon chain having a functional group selected from the group consisting of a hydroxide group, an epoxy group, a silanol group, and an alkoxysilane group and having carbon atoms of 1 to 30; hydrocarbon groups R 9, R 12 to R 14 and a hydrocarbon chain R 10 may have an element such as oxygen, nitrogen or silicon bonded in the bonding form other than the hydroxide group, the epoxy group, the silanol group, and the alkoxysilane group; and R 11 is a hydrogen or an alkyl group having carbon atoms of 1 to 8; the component has a content of the vinyl aromatic hydrocarbon of 5 to 95 % by weight; the amount of the component is 0.5 to 50 parts by weight based on 100 parts by weight of the component; and the component has an average particle size in a dispersion of 0.01 to 2 μm, wherein the modified block copolymer has a structure expressed by one of the following general formulae: (A-B) n -X, (B-A) n -X, A-(B-A) n -X, B-(A-B) n -X, X-(A-B) n -X, X-A-(B-A) n -X, X-B-(A-B) n -X, [(B-A)n] m -X, [(A-B)n] m -X, [(B-A)n-B] m -X, [(A-B)n-A] m -X where A represents a polymer block comprising primarily a vinyl aromatic hydrocarbon, and B represents a polymer block comprising primarily a conjugated diene, n is an integer of 1 or more, generally 1 to 5, and m is an integer of 2 or more, generally 2 to 10, and X is a modifying agent's residue having the functional group selected from to, and bonded to the terminals of its molecular chain. A represents a polymer block comprising primarily a vinyl aromatic hydrocarbon, and B represents a polymer block comprising primarily a conjugated diene, n is an integer of 1 or more, generally 1 to 5, and m is an integer of 2 or more, generally 2 to 10, and X is a modifying agent's residue having the functional group selected from to, and bonded to the terminals of its molecular chain.