Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will focus on the ALH* type which has a non-compact end with inhomogeneous collapsing near infinity. I will talk about a joint project with Rafe Mazzeo on the Fredholm mapping property of the Laplacian and the Dirac operator, where the geometric microlocal analysis of fibered metrics plays a central role. Application of this Fredholm theory includes the L^2 Hodge theory, polyhomogeneous expansion and local perturbation theory. I will also discuss a joint project with Yu-Shen Lin and Sidharth Soundararajan on the degeneration of such metrics which gives a partial compactification of their moduli space.