We discuss the geometry and arithmetic of the space of lattices, exploring its intricate structure and the rich mathematical concepts it embodies. We will begin by examining geodesic flats within this space, and we will investigate the diagonal action on this space, focusing on the compact orbits of this action. These orbits play a crucial role in understanding the dynamic behavior and geometric characteristics of the space. Furthermore, we will explore the profound connections between these geometric concepts and zeta functions, which encode valuable information about the spaces.