A METHOD FOR THE RING OPENING POLYMERIZATION OF UNSATURATED ALICYCLIC COMPOUNDS CONTAINING AT LEAST 4 AND NOT MORE THAN 5 CARBON ATOMS AND ONE DOUBLE BOND IN THE CYCLIC RING AND UNSATURATED ALICYCLIC COMPOUNDS CONTAINING AT LEAST 8 CARBON ATOMS AND AT LEAST ONE DOUBLE BOND IN THE CYCLIC RING IS DISCLOSED WHICH COMPRISES SUBJECTING SUCH UNSATURATED ALICYCLIC COMPOUNDS TO A CATALYST SYSTEM COMPRISING (A) A TUNGSTEN OR MOLYBDENUM HALIDE OR OXYHALIDE, (B) AT LEAST ONE LEWIS ACID OF THE FORMULA MXN WHERE M IS A METAL REPRESENTED BY ALUMINUM, ZINC, GALLIUM, TIN AND ANTIMONY; X IS A HALOGEN SUCH AS BROMINE, CHLORINE, IODINE AND FLUORINE AND N IS AN INTEGER EQUAL TO THE VALENCE OF M AND (C) A REFUCING AGENT SELECTED FROM THE GROUP CONSISTING OF METALS, METAL HYDRIDES, ALKYL SECONDARY AMINES, ALKYL TERTIARY AMINES, SATURATED ALKYL ALCOHOLS, ALKYL PHOSPHINES AND HYDROGEN.