Skein modules were introduced by Przytycki and independently by Turaev as generalizations of the polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Among these, the Kauffman bracket skein module (KBSM) has been studied most extensively. Recently, Gunningham, Jordan, and Safronov demonstrated that for any closed 3-manifold, the KBSM is finite-dimensional over $\mathbb{Q}(A)$; however, this finiteness does not extend to the KBSM over $\mathbb{Z}[A^{\pm 1}]$. Moreover, computing the KBSM of a 3-manifold remains a notoriously challenging problem, especially over this ring. In this talk, we will survey these developments and explore several open questions concerning the structure of the KBSM over $\mathbb{Z}[A^{\pm 1}]$.