Division Algebras, Extended Supersymmetries and Applications

I present here some new results which make explicit the role of the division algebras R,C,H,O in the construction and classification of, respectively, N=1,2,4,8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N=8 Malcev superaffine algebra is introduced and its relation to the non-associative N=8 SCA is discussed. A list of present and possible future applications is given.