In this talk we explain the construction of all rotational constant mean curvature (cmc) hypersurfaces of spheres by showing that every cmc hypersurface with two principal curvatures must be rotational.  Then, we explain how to compute the spectra of their Laplace and stability operators. As an example, we pick an immersed minimal 3-dimensional rotational  hypersurface of  the 4-dimensional unit sphere and show that the first two nonzero eigenvalues of the Laplacian are: approximately 0.4404 with multiplicity 1, and 3 with multiplicity 5. For this same immersion, we show that the negative eigenvalues of the Jacobi operator are: approximately -8.6534 with multiplicity 1,  -8.52078 with multiplicity 2, -3 with multiplicity 5,  -2.5596 with multiplicity 6, and  -1.17496 with multiplicity 1 (all decimal approximated). The stability index of this hypersurface is thus 15.