  A Puiseux monoid is an additive submonoid of the nonnegative rational
numbers. If $M$ is a Puiseux monoid, then the question of whether each
non-invertible element of $M$ can be written as a sum of irreducible elements
(that is, $M$ is atomic) is surprisingly difficult. Although various techniques
have been developed over the past few years to identify subclasses of Puiseux
monoids that are atomic, no general characterization of such monoids is known.
Here we survey some of the most relevant aspects related to the atomicity of
Puiseux monoids. We provide characterizations of when $M$ is finitely
generated, factorial, half-factorial, other-half-factorial, Pr\"ufer,
seminormal, root-closed, and completely integrally closed. In addition to the
atomicity, characterizations are also not known for when $M$ satisfies the
ACCP, the bounded factorization property, or the finite factorization property.
In each of these cases, we construct an infinite class of Puiseux monoids
satisfying the corresponding property.
