ecently, Oguiso addressed the following question, attributed to Gizatullin: “Which automorphisms of a smooth quartic K3 surface $D\subset\mathbb{P}^3$ are induced by Cremona transformations of the ambient space $\mathbb{P}^3$?'' When $D\subset \mathbb{P}^3$ is a smooth quartic surface, $(\mathbb{P}^3,D)$ is an example of a Calabi-Yau pair, that is, a pair $(X,D)$, consisting of a normal projective variety $X$ and an effective Weil divisor $D$ on $X$ such that $K_X+D\sim 0$. The above question is really about birational properties of the Calabi-Yau pair $(\mathbb{P}^3,D)$. In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. Then I will focus on the case of singular quartic surfaces $D\subset\mathbb{P}^3$. Our results illustrate how the appearance of increasingly worse singularities in $D$ enriches the birational geometry of the pair $(\mathbb{P}^3, D)$, and lead to interesting subgroups of the Cremona group of $\mathbb{P}^3$. This is joint work with Alessio Corti and Alex Massarenti.