From the introduction: ``Throughout this paper, let $M$ be a complete, connected and non-compact Riemannian manifold with a base point $o\in M$. We say that $M$ has an asymptotic cone if the pointed Gromov-Hausdorff limit of $((1/t)M,o)$ as $t\to\infty$ exists, and it is isometric to a Euclidean cone. The existence of asymptotic cones has been shown for manifolds with restricted sectional curvature, and then we see that the cone is generated by the Tits ideal boundary. In the case where $M$ is a Hadamard manifold, Gromov has shown that $M$ has an asymptotic cone if its Tits ideal boundary is compact. If $M$ is non-negatively curved, then $M$ has the asymptotic cone, and its Tits ideal boundary is an Aleksandrov space with curvature bounded below by 1. On the other hand, M. Gromov\ [Comment. Math. Helv. {\bf 56} (1981), no.~2, 179--195; MR0630949 (82k:53062)] and U. Abresch\ [Ann. Sci. École Norm. Sup. (4) {\bf 18} (1985), no.~4, 651--670; MR0839689 (87j:53058)] have studied manifolds of asymptotically non-negative curvature, and their topologies, and A. Kasue\ [Ann. Sci. École Norm. Sup. (4) {\bf 21} (1988), no.~4, 593--622; MR0982335 (90d:53049)] has introduced the ideal boundary of such a manifold and has given its compactification. However, G. Drees\ [Differential Geom. Appl. {\bf 4} (1994), no.~1, 77--90; MR1264910 (95e:53052)] pointed out a gap in the argument of Kasue [op. cit.] (cf. 1 and the end of 4 in [G. Drees, op. cit.]). Without smoothing the gap, Kasue's compactification is not yet completed. The main purpose of the present paper is to show the existence of asymptotic cones for a class of manifolds with restricted radial curvature. Our class includes the class of all manifolds of asymptotically non-negative curvature.''