This is more like the classical idea that the moduli problem is to express the algebraic structure naturally coming with a set (say of isomorphism classes of elliptic curves).
As Mumford pointed out in his book Geometric Invariant Theory, one might want to have the fine version, but there is a technical issue (level structure and other 'markings') that must be addressed to get a question with a chance of having such an answer.
[2] According to a recent survey by János Kollár, it "has a rich and intriguing intrinsic geometry which is related to major questions in many branches of mathematics and theoretical physics.
"[3] Braungardt has posed the question whether Belyi's theorem can be generalised to varieties of higher dimension over the field of algebraic numbers, with the formulation that they are generally birational to a finite étale covering of a moduli space of curves.
[5] It is possible to identify the coarse moduli space of special instanton bundles, in mathematical physics, with objects in the classical geometry of conics, in certain cases.