Moduli Spaces of Fano Varieties

Moduli spaces hold a pivotal position within the field of algebraic geometry. Over the past decade, the emergence and development of K-stability have significantly enriched our understanding of moduli theory for Fano varieties and log Fano pairs, giving rise to what are referred to as K-moduli spaces. The primary objective of this thesis is to delve into the realm of K-moduli spaces, specifically focusing on higher-dimensional Fano varieties and log Fano pairs. Through an in-depth exploration of these moduli spaces, we aim to harness their insights to shed light on the broader subject of moduli spaces for curves and the moduli of K3 surfaces.