Why working in compact spaces?

Question

I am trying to study moduli spaces of stable curves with n-marked points, $M_{0,n}$. However, in general the texts generally talk about the closure of this space, $\overline{M_{0,n}}$. My question, why do we sometimes prefer to work in the closure of a space rather than the normal space? Is there any specific reason for working in the closure of the moduli space of curves or maybe an example of some other space where we prefer to work on the closure of it and a reason for this? Thanks for the help!