Superconformal algebras, R symmetries and supergravities in higher dimension

Superconformal algebras embedding space-time in any dimension and signature are considered. Different real forms of the R-symmetries arise both for us ual space-time signature (one time) and for Euclidean or exotic signatures (more than one times). Application of these superalgebras are found in the context of supergravities with 32 supersymmetries, in any dimension D less than or equal to 11. These theories are related to D = 11, M, M* and M ' th eories or D = 10, IIB, IIB* theories when compactified on Lorentzian tori. All dimensionally reduced theories fall in three distinct phases specified by the number of (128 bosonic) positive and negative norm states: (n(+), n( -)) (128, 0), (64, 64), (72, 56).