  Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$,
$\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$ and
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ be the anisotropic mixed-norm Hardy space
associated with $\vec{a}$ defined via the non-tangential grand maximal
function. In this article, via first establishing a Calder\'{o}n-Zygmund
decomposition and a discrete Calder\'{o}n reproducing formula, the authors then
characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of
atoms, the Lusin area function, the Littlewood-Paley $g$-function or
$g_{\lambda}^\ast$-function. The obtained Littlewood-Paley $g$-function
characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ coincidentally
confirms a conjecture proposed by Hart et al. [Trans. Amer. Math. Soc. (2017),
DOI: 10.1090/tran/7312]. Applying the aforementioned Calder\'{o}n-Zygmund
decomposition as well as the atomic characterization of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, the authors establish a finite atomic
characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which further
induces a criterion on the boundedness of sublinear operators from
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying
this criterion, the authors obtain the boundedness of anisotropic
Calder\'{o}n-Zygmund operators from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ to
itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. The obtained atomic
characterizations of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ and boundedness of
anisotropic Calder\'{o}n-Zygmund operators on these Hardy-type spaces
positively answer two questions mentioned by Cleanthous et al. in [J. Geom.
Anal. 27 (2017), 2758-2787]. All these results are new even for the isotropic
mixed-norm Hardy spaces on $\mathbb{R}^n$.
