  Let $X$ be a ball quasi-Banach function space, $\alpha\in \mathbb{R}$ and
$q\in(0,\infty)$. In this paper, the authors first introduce the new Herz-type
Hardy spaces $\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ and
$\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ associated with ball
quasi-Banach function space $X$, via the non-tangential grand maximal function.
Then, under some mild assumptions on $X$, the authors establish the
real-variable theory for $\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$
and $\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$, in terms of maximal
function characterizations, atomic and molecular decompositions, and obtain the
boundedness of some sublinear operators from
$\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ to
$\mathcal{\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ and from
$\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ to
$\mathcal{K}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$. As appliccations, we give two
concrete function spaces which are members of Herz-type Hardy spaces associated
with ball quasi-Banach function spaces.
