  Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The
elasticity of $x$, denoted by $\rho(x)$, is the ratio of its largest
factorization length to its shortest factorization length, and it measures how
far is $x$ from having a unique factorization. The elasticity $\rho(M)$ of $M$
is the supremum of the elasticities of all non-unit elements of $M$. The monoid
$M$ has accepted elasticity if $\rho(M) = \rho(m)$ for some $m \in M$. In this
paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of
$\mathbb{Q}_{\ge 0}$). First, we characterize the Puiseux monoids $M$ having
finite elasticity and find a formula for $\rho(M)$. Then we classify the
Puiseux monoids having accepted elasticity in terms of their sets of atoms.
When $M$ is a primary Puiseux monoid, we describe the topology of the set of
elasticities of $M$, including a characterization of when $M$ is a bounded
factorization monoid. Lastly, we give an example of a Puiseux monoid that is
bifurcus (that is, every nonzero element has a factorization of length at most
$2$).
