  We prove central and non-central limit theorems for the Hermite variations of
the anisotropic fractional Brownian sheet $W^{\alpha, \beta}$ with Hurst
parameter $(\alpha, \beta) \in (0,1)^2$. When $0<\alpha \leq 1-\frac{1}{2q}$ or
$0<\beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the
renormalized Hermite variations of order $q\geq 2$, while for
$1-\frac{1}{2q}<\alpha, \beta < 1$ we prove that these variations satisfy a
non-central limit theorem. In fact, they converge to a random variable which is
the value of a two-parameter Hermite process at time $(1,1)$.
