This thesis studies a special class of Calabi-Yau metrics and singular K\"ahler-Einstein metrics that exhibit cone structures. Building on Donaldson--Sun's 2-step degeneration theory, we make progress on the precise asymptotic behavior of these metrics. The following results are proved, revealing interesting distinctions between the global and local settings.In the global setting, every complete Calabi-Yau metric with Euclidean volume growth and quadratic curvature decay, is polynomially asymptotic to its unique tangent cone at infinity. In the local setting, let $(X,p,\omega)$ be a singular K\"ahler-Einstein metric with an isolated singularity. An algebraic criterion for polynomial convergence to tangent cones is established under certain assumptions. Additionally, examples are provided where the polynomial convergence rate does not hold.

In this thesis, we also study a special class of Hermitian-Yang-Mills connections over complete non-compact K\"ahler manifolds. We introduce the notion of stability for a pair of classes that generalizes the standard slope stability. Under the assumption that both the holomorphic vector bundle and the ambient manifold can be compactified, and the K\"ahler metric satisfies certain asymptotic behavior, we prove that this new stability condition is both sufficient and necessary for the existence of projectively Hermitian-Yang-Mills metrics.