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.