  Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi
\cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture
in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not
a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral
tile. In search of lower dimensional non-spectral tiles we were led to
investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang
\cite{lagwang}. In particular, we prove here that the USC and the ``tile $\to$
spectral'' direction of Fuglede's conjecture are equivalent in any dimensions.
Also, we show by an example that the sufficient condition of Lagarias and
Szab\'o \cite{lagszab} for the existence of universal spectra is not necessary.
This fact causes considerable difficulties in producing lower dimensional
examples of tiles which have no spectra. We overcome these difficulties by
invoking some ideas of R\'ev\'esz and Farkas \cite{revfark}, and obtain
non-spectral tiles in $\RR^3$.
  Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1
and 2 dimensions. The 1 dimensional case is closely related to a number
theoretical conjecture on tilings by Coven and Meyerowitz \cite{covmey}.
