The invention relates to a reactor for performing exothermic equilibrium reactions, especially for producing methanol from synthesis gas in a multistage synthesis with intermediate condensation of the reaction product. The reactor according to the invention has a reactor shell and a multitude of series-connected and mutually fluid-connected reactor cells disposed within the reactor shell, where each of the reactor cells includes a reaction apparatus, a cooling-down apparatus and a phase separation apparatus as reactor cell elements. The reactor has a multitude of reactor planes disposed in a mutually parallel arrangement within the reactor shell, where reactor cell elements of the same kind are disposed in the same reactor plane. The inventive arrangement of the reactor cell elements enables the building of a compact reactor and reduces material stresses within the reactor by the avoidance of large temperature differences within the reactor shell.