  In the 60's, Berger famously showed that translational tilings of
$\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently,
Bhattacharya proved the decidability of translational monotilings (tilings by
translations of a single tile) in $\mathbb{Z}^2$. The decidability of
translational monotilings in higher dimensions remained unsolved. In this
paper, by combining our recently developed techniques with ideas introduced by
Aanderaa and Lewis, we finally settle this problem, achieving the
undecidability of translational monotilings of (periodic subsets of) virtually
$\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$,
where $G_0$ is a finite Abelian group. This also implies the undecidability of
translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.
