Homo- or copolymers which have terminal mercapto groups, are composed of one or more monomers from the group consisting of butadiene, isoprene and styrene and are obtained by anionic polymerization of one or more of the stated monomers in the presence of one or more organolithium compounds followed by reaction (of the living anions) with an episulfide of the general formula I ##STR1## where R is hydrogen, alkyl of 1 to 18 carbon atoms or aryl of 6 to 10 carbon atoms, with the formation of one or more primary or secondary mercapto groups at the end of the polymer chain(s) are prepared by a process in which a thiolcarboxylic acid of the general formula II ##STR2## where R.sup.1 is hydrogen, alkyl of 1 to 10 carbon atoms or --CH.sub.2 --COOH and R.sup.2 is 0 or from 1 to 10 CH.sub.2 units, is added directly to the resulting reaction mixture in an amount of from 0.5 to 2 moles of the thiolcarboxylic acid per mole of episulfide, and the reaction product is subjected to an aftertreatment