  Let $M$ be a commutative cancellative monoid, and let $R$ be an integral
domain. The question of whether the monoid ring $R[x;M]$ is atomic provided
that both $M$ and $R$ are atomic dates back to the 1980s. In 1993, Roitman gave
a negative answer to the question for $M = \mathbb{N}_0$: he constructed an
atomic integral domain $R$ such that the polynomial ring $R[x]$ is not atomic.
However, the question of whether a monoid algebra $F[x;M]$ over a field $F$ is
atomic provided that $M$ is atomic has been open since then. Here we offer a
negative answer to this question. First, we find for any infinite cardinal
$\kappa$ a torsion-free atomic monoid $M$ of rank $\kappa$ satisfying that the
monoid domain $R[x;M]$ is not atomic for any integral domain $R$. Then for
every $n \ge 2$ and for each field $F$ of finite characteristic we exhibit a
torsion-free atomic monoid of rank $n$ such that $F[x;M]$ is not atomic.
Finally, we construct a torsion-free atomic monoid $M$ of rank $1$ such that
$\mathbb{Z}_2[x;M]$ is not atomic.
