  In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in
various fields. The corresponding sign rule is determined by the `scalar
product' of the involved $\mathbb{Z}_2^n$-degrees. The
$\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the
classical one: nonzero degree even coordinates are not nilpotent, and even
(resp., odd) coordinates do not necessarily commute (resp., anticommute)
pairwise. In this article we develop the foundations of the theory: we define
$\mathbb{Z}_2^n$-supermanifolds and provide examples in the ringed space and
coordinate settings. We thus show that formal series are the appropriate
substitute for nilpotency. Moreover, the class of
$\mathbb{Z}_2^\bullet$-supermanifolds is closed with respect to the tangent and
cotangent functors. We explain that any $n$-fold vector bundle has a canonical
`superization' to a $\mathbb{Z}_2^n$-supermanifold and prove that the
fundamental theorem describing supermorphisms in terms of coordinates can be
extended to the $\mathbb{Z}_2^n$-context.
