A process for preparing 1-butene polymers, said process comprising polymerizing 1-butene or copolymerizing 1-butene with ethylene, propylene or an alpha-olefin of formula CH 2 =CHT wherein T is a C 3 -C 10 alkyl group, to obtain a copolymer having a content of comonomer derived units from 0.2% by mol to 15% by mol; in the presence of a catalyst system obtainable by contacting: a metallocene compound having the following formula wherein: M is an atom of a transition metal selected from those belonging to group 4 or to the lanthanide or actinide groups in the Periodic Table of the Elements; X, equal to or different from each other, is a hydrogen atom, a halogen atoms or R, OR, OSO 2 CF 3, OCOR, SR, NR 2 or PR 2 group, wherein R is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 1, R 2, R 5, R 6, R 7, R 8 and R 9, equal to or different from each other, are hydrogen atoms, or linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radicals, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; or R 5 and R 6, and/or R 8 and R 9 can optionally form a saturated or unsaturated, 5 or 6 membered rings, said ring can bear C 1 -C 20 alkyl radicals as substituents; with the proviso that at least one of R 6 or R 7 is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 3 and R 4, equal to or different from each other, are linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; and an alumoxane and/or compounds of formula D + E -, wherein D + is a Brønsted acid, and E - is a compatible anion. a metallocene compound having the following formula wherein: M is an atom of a transition metal selected from those belonging to group 4 or to the lanthanide or actinide groups in the Periodic Table of the Elements; X, equal to or different from each other, is a hydrogen atom, a halogen atoms or R, OR, OSO 2 CF 3, OCOR, SR, NR 2 or PR 2 group, wherein R is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 1, R 2, R 5, R 6, R 7, R 8 and R 9, equal to or different from each other, are hydrogen atoms, or linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radicals, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; or R 5 and R 6, and/or R 8 and R 9 can optionally form a saturated or unsaturated, 5 or 6 membered rings, said ring can bear C 1 -C 20 alkyl radicals as substituents; with the proviso that at least one of R 6 or R 7 is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 3 and R 4, equal to or different from each other, are linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; and M is an atom of a transition metal selected from those belonging to group 4 or to the lanthanide or actinide groups in the Periodic Table of the Elements; X, equal to or different from each other, is a hydrogen atom, a halogen atoms or R, OR, OSO 2 CF 3, OCOR, SR, NR 2 or PR 2 group, wherein R is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 1, R 2, R 5, R 6, R 7, R 8 and R 9, equal to or different from each other, are hydrogen atoms, or linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, C 3 -C 20 -cycloalkyl, C 6 -C 20 -aryl, C 7 -C 20 -alkylaryl or C 7 -C 20 -arylalkyl radicals, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; or R 5 and R 6, and/or R 8 and R 9 can optionally form a saturated or unsaturated, 5 or 6 membered rings, said ring can bear C 1 -C 20 alkyl radicals as substituents; with the proviso that at least one of R 6 or R 7 is a linear or branched, saturated or unsaturated C 1 -C 20 -alkyl radical, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R 3 and R 4, equal to or different from each other, are linear or branched, saturated or unsaturated C 1 -C 20 -alkyl, optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; and an alumoxane and/or compounds of formula D + E -, wherein D + is a Brønsted acid, and E - is a compatible anion.