A crystalline higher .alpha.-olefin polymer which is obtained from a C.sub.10 or more .alpha.-olefin and satisfies either (1) the melting point (Tm) as measured with a differential scanning calorimeter (DSC) is 20 to 100.degree. C. or (2) in an examination of spin-lattice relaxation time (T.sub.1) by solid NMR analysis, a single T.sub.1 is observed at the temperatures not lower than the melting point; and a process for producing the .alpha.-olefin polymer with a specific metallocene catalyst. The crystalline higher .alpha.-olefin polymer is excellent in low-temperature characteristics, rigidity, heat resistance, compatibility with lubricating oils, mixability with inorganic fillers, and secondary processability.