  Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this
article, the authors introduce the weak Hardy-type space $WH_X({\mathbb R}^n)$,
associated with $X$, via the radial maximal function. Assuming that the powered
Hardy--Littlewood maximal operator satisfies some Fefferman--Stein
vector-valued maximal inequality on $X$ as well as it is bounded on both the
weak ball quasi-Banach function space $WX$ and the associated space, the
authors then establish several real-variable characterizations of
$WH_X({\mathbb R}^n)$, respectively, in terms of various maximal functions,
atoms and molecules. As an application, the authors obtain the boundedness of
Calder\'on--Zygmund operators from the Hardy space $H_X({\mathbb R}^n)$ to
$WH_X({\mathbb R}^n)$, which includes the critical case. All these results are
of wide applications. Particularly, when $X:=M_q^p({\mathbb R}^n)$ (the Morrey
space), $X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space) and
$X:=(E_\Phi^q)_t({\mathbb R}^n)$ (the Orlicz-slice space), which are all ball
quasi-Banach function spaces but not quasi-Banach function spaces, all these
results are even new. Due to the generality, more applications of these results
are predictable.
