Graded Lie algebras give a uniform approach to many questions in arithmetic statistics. I'll give some background about graded Lie algebras and show how they arise in proofs about families of algebraic curves. For a certain class of graded Lie algebras, Thorne showed that we can construct families of curves as fibers of a certain categorical quotient map. I'll talk about ongoing work with Jef Laga to generalize this construction, including new examples.