Abstract: A process for the polymerization of aromatic vinyl compounds and their copolymerization with alkyl esters or nitriles of acrylic and/or methacrylic acid, wherein1. in a first stage, the monomer or monomers is or are continuously introduced at excess pressure into a first reaction zone and polymerized there to a degree of conversion of 5 - 40% by weight at a temperature of from 80.degree. to 170.degree. C;2. in a second stage, the reaction mixture is conducted at constant pressure into a second reaction zone designed for conducting highly viscous melts, the mixture flowing through this reaction zone at a rate which is 1.1 to 5 times the intrinsic rate, and polymerized there to a degree of conversion of 40 to 95% at temperatures of from 120.degree. to 250.degree. C, the excess pressure in the first and second stage being chosen so that the monomer or monomers is or are liquid, and the polymer is continuously removed from this zone with release of the pressure; and3.