I am trying to read the book 'Mirror Symmetry and Algebraic Geometry' by D. Cox and S. Katz. In the book it claims that 'the space of all complex structures on a given manifold $V$ is a well known object in algebraic geometry'. Although I have seen a (very) little bit of deformation theory, I am unfamiliar with thinking about global moduli spaces. I also saw that for curves, there is a very nice moduli stack $\mathcal{M}_{g,n}$ that is in a definite sense a global moduli space, which solves a well-defined moduli problem. But I could not find any accesible references to global moduli spaces for dimensions $\geq2$. Also it is claimed that the full complex moduli space of a Calabi-Yau manifold $V$ is a smooth manifold, but the references also had only local statements.
So what exactly is this moduli space, and in what sense is it a moduli space?