This project studies locally homogeneous geometric manifolds, which are abstract mathematical objects designed to model the physical universe. The term locally homogeneous refers to the presence of a high degree of local symmetry which is captured by an object called the (local) symmetry group. It is this symmetry group which dictates the geometry, in the following sense: the meaningful quantities we can measure in a geometric manifold, such as lengths or angles, are exactly those which are invariant under the symmetry. There are many different possible symmetry groups which lead to different types of geometric manifolds useful in many contexts across mathematics and physics. There can also be many different geometric manifolds with the same local symmetry group. These all have the same local properties but can look very different at large scale. The space of all such possibilities is called a moduli space. While the precise features, for example the shape or size, of the universe is a question for empirical physics, a moduli space is the mathematical answer to the question of what possible features the universe could have. The research goals of this project are, roughly, to better understand several special types of locally homogeneous geometric manifolds which have mysterious but tractable behavior. This project contributes to the growing base of foundational mathematical knowledge on which many innovations in science and engineering are eventually built. Broader impacts o