In computer science, graphs are one of the most common data structures used to store information. This thesis presents research on graphs designed to store other graphs as efficiently as possible. More specifically, we study different families of graphs and, for each one, we seek graphs capable of containing all those in the family while preserving their metric structure. Such graphs are called isometric universal graphs, and this thesis aims to find the smallest among them. By “smallest,” we mean those that require the least storage space. We first explore the structural and algorithmic properties of isometric universal graphs. Next, we highlight their connections to a related research area, that of distance labeling schemes. These are data structures that encode graphs while preserving their metric structure; however, unlike isometric universal graphs, these schemes are not themselves graphs. We then turn our attention to specific families of graphs. For these families, we seek to construct isometric universal graphs of the smallest size that is mathematically possible. This involves establishing lower bounds on the size of such graphs, bounds that are inherent to their structures. We are particularly interested in families of graphs with simple structures, such as trees, forests, or general graphs. Finally, we study algorithmic questions related to isometric universal graphs for these families and prove certain NP-completeness results.