We will begin by reviewing the classical moduli theory of curves developed by Deligne–Mumford and Grothendieck–Knudsen. We will then discuss its higher dimensional analogues, i.e., the KSBA moduli theory. A central issue is that flatness behaves differently in higher dimensions: it is too weak for the underlying variety, but too strong for the divisorial part. Finally, using an arrangement of lines as an example, we will illustrate how the Minimal Model Program (MMP) can be used to find stable degenerations.

Several KSBA compactifications have been constructed explicitly, with their families (all stable pairs) described in detail. Examples include pairs of plane curves with small coefficients, principally polarized abelian varieties (PPAVs), hyperplane arrangements, K3 surfaces with recognizable divisors and so on. We will briefly review some of these constructions then focusing on hyperplane arrangements, where we will discuss the two different approaches developed by Hacking–Keel–Tevelev via visible contours and by Alexeev via geometric invariant theory (GIT). Finally, we will propose several problems for graduate students to explore.