  The regularity of a Fano variety, denoted by ${\rm reg}(X)$, is the largest
dimension of the dual complex of a log Calabi--Yau structure on $X$. The
coregularity is defined to be \[ {\rm coreg}(X):= \dim X - {\rm reg}(X)-1. \]
The coregularity is the complementary dimension of the regularity. We expect
that the coregularity of a Fano variety governs, to a large extent, the
geometry of $X$. In this note, we review the history of Fano varieties, give
some examples, survey some important theorems, introduce the coregularity, and
propose several problems regarding this invariant of Fano varieties.
