  In this paper, we study the atomic structure of the family of Puiseux
monoids. Puiseux monoids are a natural generalization of numerical semigroups,
which have been actively studied since mid-nineteenth century. Unlike numerical
semigroups, the family of Puiseux monoids contains non-finitely generated
representatives. Even more interesting is that there are many Puiseux monoids
which are not even atomic. We delve into these situations, describing, in
particular, a vast collection of commutative cancellative monoids containing no
atoms. On the other hand, we find several characterization criteria which force
Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of
strongly bounded Puiseux monoids over a finite set of primes.
