A catalyst useful for polymerizing olefins, capable of providing an extremely high catalyst efficiency, excellent grain size and high bulk density of polymer; production of polymer having various molecular weight distributions from a broader range to a narrower range and controlling of die swell of formed polymer, by using a mechanical grinding and contacting means. Said catalyst shows a high activity and higher stereoregularity in the polymerization of polypropylene when it is used together with an electron donor compound.Said catalyst comprises a solid catalyst component (A) and an organometallic compound (B), said catalyst component (A) is obtained by reacting a specified organomagnesium component (i) with a specified chlorosilane compound (ii) and reacting and/or grinding-contacting the solid obtained by the reaction of (i) and (ii) with a titanium and/or vanadium compound containing at least one halogen atom.The organomagnesium component is of the formulaM.sub..alpha. Mg.sub..beta. R.sub.p.sup.1 R.sub.q.sup.2 X.sub.r Y.sub.swherein .alpha. is a number of 0 or greater than 0; .beta.>0; p, q, r and s are each a number of 0 or greater; p+q+r+s=m.alpha.+2.beta. wherein m is the valency of M; M is a metal element of the 1st to the 3rd group of the Periodical Table; R.sup.1 and R.sup.2 are the same or different hydrocarbon radicals having 1 to 20 carbon atoms; X and Y are the same or different radicals selected from the group consisting of halogen, OR.sup.3, OSiR.sup.4 R.sup.5 R.sup.6, NR.sup.7 R.sup.8 and SR.sup.9 wherein R.sup.3, R.sup.4, R.sup.5, R.sup.6, R.sup.7 and R.sup.8 are each hydrogen atom or a hydrocarbon radical having 1 to 20 carbon atoms and R.sup.9 is a hydrocarbon radical having 1 to 20 carbon atoms, and chlorosilane is of the formulaH.sub.a SiCl.sub.b R.sub.4-(a+b)wherein a and b are each a number greater than 0, having the relationships a.ltoreq.2 and a+b.ltoreq.4 wherein R is a hydrocarbon radical having 1 to 20 carbon atoms.