  Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic
tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by
translations then it admits at least one periodic tiling. Fuglede's conjecture
suggests that $\Omega$ admits an orthogonal basis of exponential functions if
and only if it tiles $\mathbb{R}^d$ by translations. Both conjectures are known
to be false in sufficiently high dimensions, with all the so-far-known
counterexamples being highly disconnected. On the other hand, both conjectures
are known to be true for convex sets. In this work we study these conjectures
for connected sets. We show that the periodic tiling conjecture, as well as
both directions of Fuglede's conjecture are false for connected sets in
sufficiently high dimensions.
