First, we will give a survey of finiteness results in arithmetic geometry of the form "the number of K-isomorphism classes of objects defined over K that are unramified over S is finite." Here, "objects" could be curves of a fixed genus at least 2 (Shafarevich's conjecture), abelian varieties, etc. Next, we will state Theorem 10.1 of Lawrence-Venkatesh, which provides a set of conditions that would imply this kind of finiteness in higher dimension. We will sketch their application of this criteria to hypersurfaces. If time permits, we will discuss some ideas involved in the proof of Theorem 10.1.