The talk will concern limiting distribution problems in the moduli space of k-dimensional lattices in $R^n$. These problems are motivated by classical lattice counting questions and serve as natural analogues of well-studied problems concerning orbits of lattice subgroups. I will present results from my recent joint work, with a particular focus on an ongoing collaboration with Nimish Shah that resolves a conjecture by U. Shapira and O. Sargent concerning orthogonal lattices of integral vectors on hyperboloids.