The periodic tiling conjecture asserts that any finite subset of a lattice Zd 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 Rd. In fact, we also obtain a counterexample in a group of the form Z2×G0 for some finite abelian 2-group G0. 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. This is joint work with Rachel Greenfeld.