A process for polymerizing 1-butene comprising the step of contacting under polymerization conditions 1-butene and optionally from 0 to 20% by mol of an alpha olefin with a catalyst system obtainable by contacting a bridged metallocene compound of formula (I) wherein R1 and R2 are an hydrocarbon radical A is a carbon atom, a germanium atom or a silicon atom; m is 1, 2; M is a metal of group 4 of the Periodic Table of the Elements; X, is hydrogen, a halogen atom, or a group R, OR, OSO2CF3, OCOR, SR, NR2 or PR2, wherein the substituents R are hydrocarbon radical; L is a moiety of formula (IIa) or (IIb) wherein T is an oxygen (O) or sulphur (S) atom or a CH2 group; and R3 and R4 are hydrogen atoms or hydrocarbon radicals; one or more alumoxanes or compounds able to form an alkylmetallocene cation; and optionally an organo aluminum compound.