An adhesive composition suitable for a hot melt adhesive or hot melt pressure-sensitive adhesive, which comprises a block copolymer (a) and a tackifier (b), wherein said block copolymer (a) has at least one polymer block A being a hydrogenated butadiene polymer block having a 1,2-bond content of 20 mol % or less and at least one polymer block B substantially having an olefin polymer structure different from the polymer block A and having a glass transition temperature of -20.degree. C. or less and a heat of crystalline fusion of 8 cal/g or lower, and containing the polymer block A before hydrogenation in an amount of 3 to 80 wt. %.