This paper provides a comprehensive overview of the theory of moduli spaces of geometric deformations, a central concept in modern geometry and theoretical physics. We explore the fundamental problem of classifying geometric structures, such as Riemannian metrics or complex structures, on a given manifold up to natural equivalence relations. The paper details the construction of moduli spaces as parameter spaces for these structures, emphasizing the role of deformation theory, group actions, and quotient constructions like Geometric Invariant Theory. Key examples, including the moduli space of Riemann surfaces (Teichmüller space) and the moduli of Calabi-Yau manifolds, are analyzed in depth. We investigate the rich geometric and topological properties of these spaces, such as the Weil-Petersson metric on Teichmüller space and the importance of compactifications like the Deligne-Mumford compactification. Furthermore, we discuss the profound connections between moduli spaces and other fields, including algebraic topology, number theory, and string theory, highlighting how the study of geometric deformations provides a unifying framework for diverse mathematical and physical phenomena. The paper synthesizes foundational results with recent advances, aiming to provide a coherent narrative on the structure, geometry, and applications of these fundamental mathematical objects.