  Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on
$\mathbb{R}^n$. In this article, via the non-tangential grand maximal function,
the authors first introduce the anisotropic mixed-norm Hardy spaces
$H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their
radial or non-tangential maximal function characterizations. Moreover, the
authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of
atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or
$g_{\lambda}^\ast$-functions via first establishing an anisotropic
Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space
$L^{\vec{p}}(\mathbb{R}^n)$. In addition, the authors also obtain the duality
between $H_A^{\vec{p}}(\mathbb{R}^n)$ and the anisotropic mixed-norm Campanato
spaces. As applications, the authors establish a criterion on the boundedness
of sublinear operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach
space. Applying this criterion, the authors then obtain the boundedness of
anisotropic convolutional $\delta$-type and non-convolutional $\beta$-order
Calder\'{o}n-Zygmund operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ to itself [or
to $L^{\vec{p}}(\mathbb{R}^n)$]. As a corollary, the boundedness of anisotropic
convolutional $\delta$-type Calder\'on-Zygmund operators on the mixed-norm
Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is
also presented.
