  Inspired by the notion of quasi-infinite divisibility (QID), we introduce and
study the class of freely quasi-infinitely divisible (FQID) distributions on
$\mathbb{R}$, i.e. distributions which admit the free L\'{e}vy-Khintchine-type
representation with signed L\'{e}vy measure. We prove several properties of the
FQID class, some of them in contrast to those of the QID class. For example, a
FQID distribution may have negative Gaussian part, and the total mass of its
signed L\'{e}vy measure may be negative. Finally, we extend the Bercovici-Pata
bijection, providing a characteristic triplet, with the L\'{e}vy measure having
nonzero negative part, which is at the same time classical and free
characteristic triplet.
