Abstract: The present invention relates to a novel catalyst used for homopolymerization of an olefin of 2 or more carbon atoms or for copolymerization of two or more olefins of 2 or more carbon atoms, as well as to a process for producing an olefin polymer using said catalyst. High molecular olefin polymers can be produced at a high yield, by homopolymerizing an olefin or copolymerizing olefins using a catalyst system consisting of a transition metal compound of the general formula M(R).sub.l (OR').sub.m X.sub.n-(l+m) and an aluminoxane or a catalyst system comprising a transition metal compound of the general formula M(R).sub.l (OR').sub.m X.sub.n-(l+m), an aluminoxane and an organic compound having at least 2 hydroxyl groups.

Abstract: A process for producing an .alpha.-olefin polymer which comprises homopolymerizing or copolymerizing an .alpha.-olefin by the use of a catalyst system comprising:(A) a solid catalyst component (A) containing at least magnesium, titanium, halogen and ester compound as indispensable components, and(B) a catalyst component (B) produced by reacting an organoaluminum compound represented by general formula R.sup.1.sub.m AlY.sub.3-m (R.sup.1 represents an aliphatic hydrocarbon group having 1 to 8 carbon atoms, Y represents halogen, hydrogen or alkoxy group having 1 to 20 carbon atoms, and m represents a number satisfying 2.ltoreq.m.ltoreq.3) with an organic silicon compound represented by general formula R.sup.2.sub.n Si(OR.sup.3).sub.4-n (R.sup.2 represents alicyclic hydrocarbon group or aromatic hydrocarbon group having 5 to 20 carbon atoms, R.sup.3 represents a hydrocarbon group having 1 to 20 carbon atoms, and n represents a number satisfying 0.ltoreq.n.ltoreq.