  In this paper, we introduce anisotropic mixed-norm Herz spaces $\dot
K_{\vec{q}, \vec{a}}^{\alpha, p}(\mathbb R^n)$ and $K_{\vec{q},
\vec{a}}^{\alpha, p}(\mathbb R^n)$ and investigate some basic properties of
those spaces. Furthermore, establishing the Rubio de Francia extrapolation
theory, which resolves the boundedness problems of Calder\'on-Zygmund operators
and fractional integral operator and their commutators, on the space $\dot
K_{\vec{q}, \vec{a}}^{\alpha, p}(\mathbb R^n)$ and the space $K_{\vec{q},
\vec{a}}^{\alpha, p}(\mathbb R^n)$. Especially, the Littlewood-Paley
characterizations of anisotropic mixed-norm Herz spaces also are gained. As the
generalization of anisotropic mixed-norm Herz spaces, we introduce anisotropic
mixed-norm Herz-Hardy spaces $H\dot K_{\vec{q}, \vec{a}}^{\alpha, p}(\mathbb
R^n)$ and $HK_{\vec{q}, \vec{a}}^{\alpha, p}(\mathbb R^n)$, on which atomic
decomposition and molecular decomposition are obtained. Moreover, we gain the
boundedness of classical Calder\'on-Zygmund operators.
