  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, the authors give the dual space of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which was asked by Cleanthous et al. in
[J. Geom. Anal. 27 (2017), 2758-2787]. More precisely, via first introducing
the anisotropic mixed-norm Campanato space
$\mathcal{L}_{\vec{p},\,q,\,s}^{\vec{a}}(\mathbb{R}^n)$ with $q\in[1,\infty]$
and $s\in\mathbb{Z}_+:=\{0,1,\ldots\}$, and applying the known atomic and
finite atomic characterizations of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, the
authors prove that the dual space of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ is
the space $\mathcal{L}_{\vec{p},\,r',\,s}^{\vec{a}}(\mathbb{R}^n)$ with
$\vec{p}\in(0,1]^n$, $r\in(1,\infty]$, $1/r+1/r'=1$ and
$s\in[\lfloor\frac{\nu}{a_-}(\frac{1}{p_-}-1) \rfloor,\infty)\cap\mathbb{Z}_+$,
where $\nu:=a_1+\cdots+a_n$, $a_-:=\min\{a_1,\ldots,a_n\}$,
$p_-:=\min\{p_1,\ldots,p_n\}$ and, for any $t\in \mathbb{R}$, $\lfloor
t\rfloor$ denotes the largest integer not greater than $t$. This duality result
is new even for the isotropic mixed-norm Hardy spaces on $\mathbb{R}^n$.
