  In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n \geq 2$, do
appear quite frequently. The corresponding sign rules are determined by the
`scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The present paper is
the first of a series on $\mathbb{Z}_2^n$-Supergeometry. The new theory
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 (the parity is the parity of
the total degree). It is based on the hierarchy: `
$\mathbb{Z}_2^0$-Supergeometry (classical differential Geometry) contains the
germ of $\mathbb{Z}_2^1$-Supergeometry (standard Supergeometry), which in turn
contains the sprout of $\mathbb{Z}_2^2$-Supergeometry, etc.' The
$\mathbb{Z}_2^n$-supergeometric viewpoint provides deeper insight and
simplified solutions; interesting relations with Quantum Field Theory and
Quantum Mechanics are expected. In this article, we define
$\mathbb{Z}_2^n$-supermanifolds and provide examples in the atlas, the ringed
space and coordinate settings. We thus show that formal series are the
appropriate substitute for nilpotency. Moreover, the category of
$\mathbb{Z}_2^n$-supermanifolds is closed with respect to the tangent and
cotangent functors. The fundamental theorem describing supermorphisms in terms
of coordinates is extended to the $\mathbb{Z}_2^n$-context.
