Thermoplastic, branched aromatic polyphosphonates with average molecular weights (number average M.sub.n) of from 11,000 to more than 200,000, obtainable by transesterifying (a) from 97 to 100 moles of at least one diaryl phosphonate, and (b) from 0 to 3 moles of at least one triaryl phosphate--the sum of the moled of (a) and (b) being 100--with (c) from 90 to 99 moles of at least one aromatic dihydroxy compound, and (d) from 0 to 3 moles of at least one aromatic trihydroxy or tetrahydroxy compound--with the proviso that at least one of the components (b) and (d) is present in a quantity of at least 0.001 mole and that, where (b) and (d) are used, the maximum limit on the moles is 3.0.