Abstract: This invention relates to a catalytic component for the polymerization of .alpha.-olefins and provides a catalytic component of magnesium support type capable of exhibiting a high stereoregularity while maintaining an improved strength as well as a high catalytic activity. The features thereof consist in a catalyst component for the polymerization of .alpha.-olefins obtained by contacting (A) a solid component comprising, as essential components, magnesium, titanium, a halogen and an electron-donating compound with (D) an olefin in the presence of (B) a trialkylaluminum and (C) a dimethoxy group-containing compound represented by the general formula R.sup.1 R.sup.2 Si(OCH.sub.3).sub.2 where R.sup.1 and R.sup.2 are, same or different, aliphatic hydrocarbon groups with 1 to 10 carbon atoms and having a volume, calculated by the quantum chemistry calculation, of 230 to 500 .ANG..sup.3 and an electron density of oxygen atoms in the methoxy group, calculated similarly, ranging from 0.685 to 0.800 A. U.