A process for obtaining 1-butene polymers comprising the step of contacting under polymerization conditions 1-butene and optionally ethylene, propylene or said alpha-olefin, in the presence of a catalyst system obtainable by contacting: a) a metallocene compound of formula (I): ##STR00001## wherein: M is an atom of a transition metal belonging to group 3, 4, 5, 6 or to the lanthanide or actinide groups in the Periodic Table of the Elements; p is an integer from 0 to 3; X, is a hydrogen atom, a halogen atom, or a hydrocarbon group; R.sup.1 is a hydrocarbon group; R.sup.2, R.sup.3 and R.sup.6 are hydrogen atoms or hydrocarbon groups; R.sup.4 and R.sup.5 join to form a condensed saturated or unsaturaded 4 7 membered ring; and L is a divalent bridging group; and b) an alumoxane or a compound able to form an alkylmetallocene cation.