Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Among these, there exists the notion of volume preserving maps. A natural and tough problem is classifying log Calabi-Yau pairs up to volume preserving equivalence. For this task, one can make use of the coregularity, the most important volume preserving invariant of a log Calabi-Yau pair. Recently, other invariants and refinements in the classification have been studied, especially in the coregularity 0 case. In this talk, I will survey some results, applications, and comment on joint works regarding what is known to date about the classification for the case where the ambient variety is a nonsingular Fano threefold.