  Roughly speaking, $\mathbb{Z}_2^n$-manifolds are `manifolds' equipped with
$\mathbb{Z}_2^n$-graded commutative coordinates with the sign rule being
determined by the scalar product of their $\mathbb{Z}_2^n$-degrees. We examine
the notion of a symplectic $\mathbb{Z}_2^n$-manifold, i.e., a
$\mathbb{Z}_2^n$-manifold equipped with a symplectic two-form that may carry
non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and results of
symplectic geometry generalise to the `higher graded' setting, including a
generalisation of Darboux's theorem.
