  This paper focuses on the undecidability of translational tiling of
$n$-dimensional space $\mathbb{Z}^n$ with a set of $k$ tiles. It is known that
tiling $\mathbb{Z}^2$ with translated copies with a set of $8$ tiles is
undecidable. Greenfeld and Tao gave strong evidence in a series of works that
for sufficiently large dimension $n$, the translational tiling problem for
$\mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the
undecidability of translational tiling of $\mathbb{Z}^3$ with a set of $6$
tiles.
