Abstract: Polyamides, oligomers thereof or mixtures thereof, if required with further reaction products, are prepared by reacting aminonitriles or dinitriles and diamines or a mixture containing aminonitrile, dinitrile and diamine, and, if required, further polyamide-forming monomers and/oligomers with water in a reactor (1) having a vertically oriented longitudinal axis, by a process in which, in the reactor (1), the reaction product is removed from the bottom and ammonia formed and any further low molecular weight compounds formed and water are taken off via the top (2), wherein the reactor (1) has at least two chambers (4) arranged one on top of the other in the longitudinal direction, the chambers (4) being separated from one another by liquid-tight trays (5), each chamber (4) being connected by a liquid overflow (6) to the chamber (4) directly underneath and a liquid product stream being taken off by the liquid overflow (6) of the lowermost chamber (4), the gas space (7) above the liquid level in each chambe