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.