This paper presents a numerical study of the packing of nonspherical particles by the use of the discrete element method. The shapes considered are oblate and prolate spheroids, with the aspect ratio varying from 0.1 to 7.0. It is shown that the predicted relationship between packing fraction and aspect ratio is consistent with those reported in the literature. Ellipsoids can pack more densely than spheres. The maximum packing fraction occurs at an aspect ratio of 0.6 for oblate spheroids, and 1.80 for prolate spheroids. The packing characteristics with aspect ratio are further analyzed in terms of structural parameters such as coordination number and radial distribution function. It is shown that ellipsoids with small or large aspect ratios tend to give a locally ordered structure. The results demonstrate that DEM provides a useful method to investigate the packing dynamics of ellipsoidal particles.