A reactor is described for the production of C2 to C8 olefins from gaseous oxygenate and H2O and one or more material flows containing C2 C4, C5, C6, C7, C8 olefin and paraffin at 400° to 470° C., wherein several reaction stages which the material flow can pass through from the top to the bottom, each consisting of a support base with a catalyst layer situated on it, are arranged in a closed, upright container. In order to be able in each case to lower the temperature of the reaction mixture leaving the reaction stages before it enters into the next reaction stage, it is provided that each support base consists of cells which are placed closely next to each other with no gaps and which are securely attached to each other and filled with catalyst, and in the space formed by two neighboring reaction stages, respectively, an assembly of nozzle tubes is installed for spraying a liquid phase containing H2O and DME and/or MEOH, using a water-saturated gas phase containing mainly DME and/or MEOH, in the direction of the following reaction stage downstream.