There are disclosed substantially linear polyarylene ether sulfones having a reduced viscosity of having a reduced viscosity of ca. 0.25 to ca. 1.5 dl/g (measured in a 1% solution in DMF at 25.degree. C.) and consisting essentially of 95-99.8% by weight of segments of formula I and of 5-0.2% by weight of segments of formula IIa, formula IIb and/or formula IIc ##STR1## the percentages by weight being based on the entire polymer and the molecular weight (number average) of the segments of formula I being 6000 to ca. 60 000, if the polymer contains solely structural units of formula IIa and/or IIb, and 1000 to ca. 60 000 if the polymer contains more than 0.1% by weight of structural units of formula IIc, and R consists of the recurring structural units of formula Ia and/or of up to 99.8% by weight, based on the total polymer, of recurring structural units of formula Ib ##STR2## wherein R.sub.1 is C.sub.1 -C.sub.6 alkyl, C.sub.3 -C.sub.10 alkenyl, phenyl or halogen, p is an integer from 0 to 4 Ar.sub.1 and Ar.sub.2 are each independently of the other divalent carbocylic-aromatic radicals, Z.sub.1 is a divalent radical of a cycloaliphatic, aromatic or araliphatic dihydroxy compound after removal of both hydroxyl groups, Z.sub.2 is a divalent radical of a cycloaliphatic, aromatic or araliphatic di-secondary amino compound after removal of both N-hydrogen atoms, m is an integer from 1 to ca. 10, and Z.sub.3 is a trivalent radical of a cycloaliphatic, aromatic or araliphatic compound containing hydroxy and/or amino groups after removal of the hydroxyl groups and/or active hydrogen atoms bound to amino nitrogen atoms, which radicals Ar.sub.1, Ar.sub.2, Z.sub.1, Z.sub.2 and Z.sub.3 may be substituted by one to four C.sub.1 -C.sub.6 alkyl groups, C.sub.3 -C.sub.10 alkenyl groups, phenyl groups or halogen atoms, and in which radicals Z.sub.1, Z.sub.2 and Z.sub.3 one, two, three or four ring carbon atoms may be replaced by oxygen, sulfur and/or nitrogen atoms.