  Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function
satisfying the globally log-H\"older continuous condition. In this article, the
authors first introduce the variable weak Hardy space on $\mathbb R^n$,
$W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then
establish its radial or non-tangential maximal function characterizations.
Moreover, the authors also obtain various equivalent characterizations of
$W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the
Lusin area function, the Littlewood-Paley $g$-function or
$g_{\lambda}^\ast$-function. As an application, the authors establish the
boundedness of convolutional $\delta$-type and non-convolutional $\gamma$-order
Calder\'on-Zygmund operators from $H^{p(\cdot)}(\mathbb R^n)$ to
$W\!H^{p(\cdot)}(\mathbb R^n)$ including the critical case
$p_-={n}/{(n+\delta)}$, where $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$
