This invention relates to thermoplastically deformable branched aliphatic polyamides prepared by the hydrolytic polycondensation of lactams in the presence of small quantities of .alpha.-amino-.epsilon.-caprolactam (ACL) and approximately equivalent quantities of polycarboxylic acids (preferably dicarboxylic acids) and to a process for their preparation which is distinguished by considerably reduced polycondensation times without giving rise to cross-linked products. The process optionally in addition employs a solid phase after-condensation in which rapid further condensation takes place to the products with increased melt viscosity which are branched but are still soluble in m-cresol.