The talk will discuss two aspects of gravitational instantons: mode stability of gravitational instantons, and T²-symmetric (toric) gravitational instantons. First, I will discuss Ricci-flat perturbations of gravitational instantons of Petrov type D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme spin weight satisfy a Riemannian version of the separable Teukolsky equation. As a step toward infinitesimal rigidity of the type D Kerr and Taub-bolt families of instantons, I will discuss the main ideas behind the proof of mode stability, that is, the fact that the Teukolsky equation admits no solutions compatible with regularity and asymptotic (local) flatness. After this, I will turn to instantons with toric symmetry. For an asymptotically locally Euclidean (ALE) or ALF gravitational instanton (M, g) with toric symmetry, the signature of (M, g) can be expressed directly in terms of its rod structure. Applying Hitchin?Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, one can formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. I will give a brief outline of these results, and then discuss the special case of rod structures with three turning points.