The present invention provides a block 
copolymer or a hydrogenated product thereof excellent in low-temperature shrinkability, natural shrinkability, rigidity and the like, excellent in a balance of physical properties such as blocking resistance, resistance to fusion bonding in hot water, 
impact resistance and the like, and having a few fish eyes (FE's) caused by gels. Further, the invention provides a heat shrinkable film and a heat shrinkable multilayer film suitable for drink container packaging, cap seals and the like, using such a block 
copolymer or the hydrogenated product thereof. The invention provides a block 
copolymer having a weight ratio of a vinyl 
aromatic hydrocarbon and a 
conjugated diene of 60 / 40 to 90 / 10 and a number average molecular weight measured by 
gel permeation chromatography (GPC) of 30,000 to 500,000, wherein the vinyl 
aromatic hydrocarbon constituting the block copolymer has a block rate of from 10 to 90% by weight, the vinyl 
aromatic hydrocarbon polymer blocks constituting the block copolymer have a peak molecular weight within the molecular 
weight range of 5,000 to 30,000, and 40 to 80% by weight of the vinyl aromatic 
hydrocarbon polymer blocks have a molecular weight of 35,000 or less.