A catalyst for the polymerization of olefins which comprises:(A) a solid catalyst component prepared by bringing a co-ground product of a substance (a) obtained by reacting magnesium with at least two molar proportion of an alkyl monohalide in the absence of a solvent but in the presence of iodine and a dialkyl phthalate (b) and optionally existing titanium tetrachloride (c) into contact with titanium tetrachloride in the presence or absence of an aromatic hydrocarbon and/or halohydrocarbon.(B) a 2,6-dialkyl- or 2,2,6,6-tetraalkyl-piperidine derivative, and(C) an organoaluminum compound selected from a trialkyl aluminum, a dialkylaluminum halide and an alkylaluminum dihalide, as well as the solid catalyst component (A) for the catalyst. This catalyst characterized by the use of the specific solid catalyst component exhibits a high polymerization activity combined with stereospecific performance over a prolonged polymerization time and is free from any problem of corrosion and odor in the resultant polymer.