  We exhibit a subset of a finite Abelian group, which tiles the group by
translation, and such that its tiling complements do not have a common spectrum
(orthogonal basis for their $L^2$ space consisting of group characters). This
disproves the Universal Spectrum Conjecture of Lagarias and Wang. Further, we
construct a set in some finite Abelian group, which tiles the group but has no
spectrum. We extend this last example to the groups $\ZZ^d$ and $\RR^d$ (for $d
\ge 5$) thus disproving one direction of the Spectral Set Conjecture of
Fuglede. The other direction was recently disproved by Tao.
