In 2004, it is proved that the cut locus of a general point on two dimensional ellipsoid is a a segment of a curvature line and proved Jacobi's last geometric statement on the singularities of the conjugate locus. We will show the cut loci of a general point on higher dimensional ellipsoids are closed disks of codimension one and determine the singularities of the conjugate loci. Moreover we will discuss other related topics.