  In commutative monoids, the $\omega$-value measures how far an element is
from being prime. This invariant, which is important in understanding the
factorization theory of monoids, has been the focus of much recent study. This
paper provides detailed examples and an overview of known results on
$\omega$-primality, including several recent and surprising contributions in
the setting of numerical monoids. As many questions related to
$\omega$-primality remain, we provide a list of open problems accessible to
advanced undergraduate students and beginning graduate students.
