The terminally functionalized polymers are prepared from living polymers obtained by anionic polymerization of olefinically unsaturated monomers by reacting the living polymers first with capping reagents comprising nitrogen compounds of the general formulae I to VII: ##STR1## where R.sup.1 and R.sup.4 are each H, alkyl cycloalkyl or aryl, R.sup.2, R.sup.3 and R.sup.5 are each alkyl, cycloalkyl or aryl and A is a substituted or unsubstituted bridge which, in addition to the C members, can also contain N or O members and ##STR2## where R.sup.6 and R.sup.7 are each alkyl or aryl, A is a single-membered substituted or unsubstituted carbon bridge and B is a 2- to 12-membered bridge of which at least those members adjacent to the nitrogen atom comprise --C(R.sup.8, R.sup.9) radicals, where R.sup.8 and R.sup.9 can be identical or different and are H, alkyl, cycloalkyl or aryl, and the nonadjacent members can be not only --C(R.sup.8, R.sup.9) radicals but also ethyl or N-alkyl or N-aryl-imino groups, and the polymer obtained is then made to react with a terminating agent which contains halogen or acid anhydride groups, suitable capping reagents being compounds of the class of 1,5-diazabicyclo[3,1,0]hexane and suitable terminating agents being chloromethylstyrenes, acryloyl chloride, methacryloyl chloride, epichlorohydrin, acrylic anhydride, methacrylic anhydride or maleic anhydride.