  We develop a framework that allows one to describe the birational geometry of
Calabi-Yau pairs $(X,D)$. After establishing some general results for
Calabi-Yau pairs $(X,D)$ with mild singularities, we focus on the special case
when $X=\mathbb{P}^3$ and $D\subset \mathbb{P}^3$ is a quartic surface. We
investigate how the appearance of increasingly worse singularities on $D$
enriches the birational geometry of the pair $(\mathbb{P}^3, D)$.
