In the late 70s, V. Rittenberg and D. Wyler introduced multi-graded (also called color) algebras. These are generalizations of the notion of superalgebra, in which the $\mathbb{Z}_2$-grading is replaced by $\mathbb{Z}_2^n$-gradings, $n>1$, and the considered sign rule is modified accordingly (i.e. is determined by the scalar product of the involved $\mathbb{Z}_2^n$-degrees). The development of $\mathbb{Z}_2^n$-superalgebra theory, and the corresponding geometry, recently acquired new relevance: V. Ovsienko and S. Morier-Genoud observed that a remarkable class of classical non-commutative algebras, namely quaternions and more generally Clifford algebras, are in fact $\mathbb{Z}_2^n$-commutative for some $n$. In this talk, I will report on joint works with N. Poncin and J. Grabowski, concerning a first study of the notion of manifold in this multigraded setting. Contrary to a widespread belief, this $\mathbb{Z}_2^n$-supergeometry present important differences with the classical supergeometry,  and provides a sharpened viewpoint.