  The periodic tiling conjecture asserts that any finite subset of a lattice
$\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles
periodically. In this work we disprove this conjecture for sufficiently large
$d$, which also implies a disproof of the corresponding conjecture for
Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a
group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group
$G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other
non-horizontal lines are constrained to lie in a certain class of "$2$-adically
structured functions", in terms of certain functional equations that can be
encoded in turn as a single tiling equation, and then demonstrating that
solutions to this Sudoku puzzle exist but are all non-periodic.
