There are disclosed a polycyclic norbornene or its derivative hydrogenated copolymer which containsa repeating unit represented by the following formula [I] or the alkyl-substituted derivative thereof (A),a repeating unit represented by the following formula [II] or the alkyl-substituted derivative thereof (B) and, as desired,5-85% by mole of at least one repeating unit (C) selected from a repeating unit represented by the following formula [III], the alkyl-substituted derivative thereof, a repeating unit represented by the following formula [IV], the alkyl-substituted derivative and the alkylidene-substituted derivative thereof, and a process for producing the polycyclic norbornene or its derivative ring-opening hydrogenated copolymer ##STR1## wherein ------ represents either a single bond or a double bond.