For fixed degree $d$, one could ask for a meaningful compactification of the moduli space of smooth degree $d$ surfaces in $\mathbb{P}^3$. In other words, one could ask for a parameter space whose interior points correspond to [isomorphism classes of] smooth surfaces and whose boundary points correspond to degenerations of these surfaces. Motivated by Hacking's work for plane curves, I will discuss a KSBA compactification of this space by considering a surface $S$ in $\mathbb{P}^3$ as a pair $(\mathbb{P}^3, S)$ satisfying certain properties. We will study an enlarged class of these pairs, including singular degenerations of both $S$ and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree $d$ is odd, we can give a rough classification of the objects on the boundary of the moduli space.