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.