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We review the novel quasiconformal realizations of exceptional U-dualitygroups whose "quantization" lead directly to their minimal unitary irreduciblerepresentations. The group $E_{8(8)}$ can be realized as a quasiconformal groupin the 57 dimensional charge-entropy space of BPS black hole solutions ofmaximal N=8 supergravity in four dimensions and leaves invariant "lightlikeseparations" with respect to a quartic norm. Similarly $E_{7(7)}$ acts as aconformal group in the 27 dimensional charge space of BPS black hole solutionsin five dimensional N=8 supergravity and leaves invariant "lightlikeseparations" with respect to a cubic norm. For the exceptional N=2Maxwell-Einstein supergravity theory the corresponding quasiconformal andconformal groups are $E_{8(-24)}$ and $E_{7(-25)}$, respectively. Theseconformal and quasiconformal groups act as spectrum generating symmetry groupsin five and four dimensions and are isomorphic to the U-duality groups of thecorresponding supergravity theories in four and three dimensions, respectively.Hence the spectra of these theories are expected to form unitaryrepresentations of these groups whose minimal unitary realizations are alsoreviewed.Comment: Invited talk at the first Gunnar Nordstroem Symposium on Theoretical  Physics (Helsinki, Aug. 2003

REALIZATIONS OF EXCEPTIONAL U-DUALITY GROUPS AS CONFORMAL AND QUASI-CONFORMAL GROUPS AND THEIR MINIMAL UNITARY REPRESENTATIONS