The invention has an object to provide a catalyst for polymerizing or copolymerizing an .alpha.-olefin, catalyst constituent thereof, and method of polymerizing .alpha.-olefins with the catalyst, for production of .alpha.-olefin polymers or copolymers with high hydrogen response, high polymarization reaction rate, high stereoregularity and excellent melt fluidity. The invention discloses a catalyst constituent of the catalyst for polymerizing or copolymerizing an .alpha.-olefin, represented by Formula 37 or 38: Si(OR.sup.1).sub.3(NR.sup.2R.sup.3)Formula 37 (where in Formula 37, R.sup.1 is a hydrocarbon group with 1 to 6 carbon atoms; R.sup.2 is a hydrocarbon group with 1 to 12 carbon atoms or hydrogen; and R.sup.3 is a hydrocarbon group with 1 to 12 carbon atoms) RNSi(OR.sup.1).sub.3Formula 38 (where in Formula 38, R.sup.1 is a hydrocarbon group with 1 to 6 carbon atoms; and RN is a cyclicl amino group).