Hyperkähler manifolds are one of the simplest examples of Einstein manifolds. They are Ricci-flat Riemannian manifolds with special holonomy. In dimension 4, hyperkähler 4-manifolds can be purely described by a triple of symplectic 2-forms that satisfy the pointwise orthonormal condition with respect to the wedge product.

In this dissertation, we proved the compactness of a set of hyperkähler 4-manifolds with boundary under Cheeger-Gromov topology, where we assume only geometric control on the boundary and topological conditions. We showed that our proof can be extended to Einstein 4-manifolds with boundary by assuming only additional topological conditions.

Furthermore, we discuss the period map for K3 surfaces in a differential geometric setting. We gave a simple proof for the surjectivity of the period map, without invoking Yau's theorem on the Calabi conjecture or any algebraic geometry. The key is to show that when a sequence of hyperkähler metrics has bounded period in some sense, then the sequence has a convergent subsequence under Cheeger-Gromov topology.

Toric manifolds and toric varieties have played an important and particularly illuminating role in algebraic, symplectic, and K\"ahler geometry going back at least to the 1970's. The high degree of symmetry in many cases allows one to reduce the complexity of a given geometric question, giving the impression that we can really ``see'' the structure. A notable example of this in the K\"ahler setting is Donaldson's classic paper [27].

Until relatively recently, this has been mostly confined to the setting of compact manifolds (complete varieties). Both the algebraic and the symplectic perspectives have come to be fairly well understood individually in the non-compact case, but there has been little application of the intersection of the two theories, which naturally comprises toric K\"ahler geometry. The notable exception to this rule is the case of toric K\"ahler cones [56, 55, 39, 17, 16]. These non-compact manifolds inherit a great deal of structure from their compact \emph{link}, which are interesting spaces in their own right. However, these manifolds are (typically) highly singular at precisely one point.

In this dissertation, we describe a procedure for the study of K\"ahler geometry on smooth non-compact toric manifolds. This involves tying together the algebraic and symplectic perspectives. We study a class of toric manifolds and show that here we can apply a non-compact version of the classical Delzant classification together with its application to K\"ahler geometry [4, 41, 24, 42]. We apply this to non-compact shrinking K\"ahler-Ricci solitons, where we propose a useful class of complete K\"ahler metrics, and ultimately prove a general uniqueness theorem for shrinking K\"ahler-Ricci solitons on non-compact toric manifolds.