the eigenvalues of the laplacian encode fundamental geometric information about a riemannian metric. as an example of their importance, i will discuss how they arose in work of cao, hamilton and illmanan, together with joint work with stuart hall, concerning stability of einstein manifolds and ricci solitons. i will outline progress on these problems for einstein metrics with large symmetry groups. we calculate bounds on the first non-zero eigenvalue for certain hermitian-einstein four manifolds. similar ideas allow us estimate to the spectral gap (the distance between the first and second non-zero eigenvalues) for any toric kaehler-einstein manifold m in terms of the polytope associated to m. i will finish by discussing a numerical proof of the instability of the chen-lebrun-weber metric.