  We review the novel quasiconformal realizations of exceptional U-duality
groups whose "quantization" lead directly to their minimal unitary irreducible
representations. The group $E_{8(8)}$ can be realized as a quasiconformal group
in the 57 dimensional charge-entropy space of BPS black hole solutions of
maximal N=8 supergravity in four dimensions and leaves invariant "lightlike
separations" with respect to a quartic norm. Similarly $E_{7(7)}$ acts as a
conformal group in the 27 dimensional charge space of BPS black hole solutions
in five dimensional N=8 supergravity and leaves invariant "lightlike
separations" with respect to a cubic norm. For the exceptional N=2
Maxwell-Einstein supergravity theory the corresponding quasiconformal and
conformal groups are $E_{8(-24)}$ and $E_{7(-25)}$, respectively. These
conformal and quasiconformal groups act as spectrum generating symmetry groups
in five and four dimensions and are isomorphic to the U-duality groups of the
corresponding supergravity theories in four and three dimensions, respectively.
Hence the spectra of these theories are expected to form unitary
representations of these groups whose minimal unitary realizations are also
reviewed.
